%!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: golden.dvi %%Pages: 48 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMR17 CMMI12 CMSY10 CMR12 CMR10 CMMI10 CMBX10 CMSY8 %%+ EUFM10 CMBX12 MSBM10 CMMI8 CMR8 EUFM7 CMTI12 CMR6 LASY10 CMSY6 %%+ CMEX10 CMMI6 MSBM7 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips golden.dvi -o %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2003.03.04:2218 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: texps.pro %! 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /parenleftbig put dup 1 /parenrightbig put dup 12 /vextendsingle put dup 13 /vextenddouble put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 40 /braceleftBigg put dup 56 /bracelefttp put dup 58 /braceleftbt put dup 60 /braceleftmid put dup 82 /integraltext put dup 83 /uniontext put dup 88 /summationdisplay put dup 90 /integraldisplay put dup 112 /radicalbig put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueXX 5000774 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6d5649664203ab24acb938d58d246707ffe7 d62f04bec4b70c21ef75beb2b812622b3c74e969d72d3cd11bd7106294a99caf 0b1629bc7d4de6b96ca82930831d64575f23f4ad06a0e45e315b1d392411be8d 6d73c998789ff258a07a3c8c2057325784514c845500bfd1a971310cfc11d41c 1a167dbd5ff012c60add4e87325f6e5299032a839de65fb1473a166aae1876a4 414a434f22c1d241591fb36f857df6fa930608750ffc0c54f44994662b1f00f1 400bf752ea8d83ffc4cb77a290bc2d99981ae59a191748ba5c7ba1a9d2583fd2 1398452b6ff5d83a059f7eadcd2ef744e9dd22bdf9c79d049bf06835e878c32b 7765c69bdd8ef4deb4ea7cfff4cf9354a4ddffa689de961d16772491c7afbd7f ffde42400764c68e954ee5c455a5687959829bc3b319b2147deaab3628662c80 30c5e02fea09609abe4eaa12e217bc3af673f1bc36a7039eb13fcacb4218fe0f c5a3f9452d4edf46cc91db67b624d4f2d37502fb9f11af4da18ca40b61097f95 d44329375467ed13c5cb585ec53f62b83ef9502cc755af44bf32b87b8ae9f3f2 f8dbf72dab90acafbacd280db6aaffaefdff6d5eff26669bac56280a950560e3 d01714ba38659b0e4e28bf8cdc1f15ff7e3ea9d95a6afbbe3d8e6905aa6d531d 6b6dcab89e2036cc492c65f76ddcc3cfcff33fde3d9ffd10f7a90530fd5c7683 4329baa902e701ca42717476298c9ff511b1941244bb1dfd0e5cab92a43970cd 1bdb292eadd055c97972844fdcdb0aa06401e7b2632b9d134f96bce0064860f3 9859da7a9a82be0a840672a1331212752ae1b521b693b384a72dba8d37a86eaf a0341e7fc5f6ef7a8c566aacb91e8bf5005c18837894c4eacff41dba54e071ca 83fd0039712673a39043dbf95fb0fdf6a32db8077957395cacedb1e92bbe3ae3 de1f21617a1e25c98eb248bf37a7ab7c633061d9ad392364039a9e87ca851c5f 37cdabcc642702e291c5454c1b5dec5a20d371a0ba899451ca118715308a296f d82043b674c21b504d40f7f3a24aba3b909f81212e5b666618732e169a624351 67e81afe747d9a2cc5f2c25eb7ba442c2ec0cdb29a1f591c31c9f3541e43b575 da40327509fedd183008a0996b3f376da4f894de83333938477bfe7dc3804a65 59fb5756d24d696b099882eb5934155990536dca6615f67adaddd295cf357b75 11aa6139fe50ce13d8ef9784ba5873f464133f8fefbac7dbefecbad21fbae8c3 cd25c04d92a206e8b2243d5639397346f36e06a92a14d78487810123d80826ef 34ac5ff441d5e3c806c955ae1666fc595b7829b5b1fde24e359367673ad1ec4a 5136f5e9b40bcdf66ff865f6b41b2c2fc9b939325399d967dd2efb56538d4279 8b43366893c661a2ebeb58f8cbd312a0fe513942ae1b513152d56b053209963c 731a79505eee0e4489d519db9176ce0da77b460b97c987124a03e33cd9d0b393 d931e86bc7dc08c3d08806618a9c5f980a3aed81a4632738e76cad97e618a9de ac86a22aebace24dd49fc8856c8a30833f374eab04ac1088b294495988ea9265 2fd48991ace4038267eae1c51fe31a986153883d5194ad45e0336be3c359a927 026045cf38d02442bf70f1ea284f78fa815198139f37f1dcb727e54774c6f0f7 d2949f5c4ae37c7484783bff44241fa24bd4efc4ba37b7a75af2dd846733cc76 17daf7eba7cc4f0836926f935921da66fb87699075901ebfa4c2087ba9946898 6588f73d79fa533a5a804cb1818c8d00d8eba4143c8db22695682d267d93207e 2dfd53d038a0af11bc23718f26c41b5cb4a9293412936ef6bf1ad41ebf44dee0 d51e8e960fcd51bcac83abf5aff2c82179eefe236e27496725a9a8b34aab7879 0b766d0ae8bff7f7448985d7b98c47afb4382d527f8b0527b3618a5979ff1a3e c65bf72c1be7f022371143de39ab8312122504da0bc87741b9e2c398076dd31c 1e7e6a4dab871a92a39950e7d0c9ab04064385e55370ebe8925c6c310d6f6514 52837ed3f8542a0b9abeeb28a4195c10c181ae35f4e9110c4d00b63dd0113bcb 8274cb5f4d9fcf0034c0961fd3d716a78eb8230f8b6e15b2f80b6f74d0480275 24d49cfc48061f90d4f2a7cf27b35035ef9faea7d2e721480041e0095fe21a8d 9e0dd4d15eef36a05bc8592338939f5179d81522c9fa141da2631d9eac5ed8df cde25ff53bc2c3b38f1a48f3ffb9e59a24378cad17ff1c938df62af5e8123870 2ccc3d0b8e8be39f8a4a869b41781aa6593c18d2f38d927450565ee166f0e4a0 fba673632d214b6cf7f1ea0e6d4be5fd3ac22dd4cb8cdf9de62c66f4c95cd61c 46a7ac05e52ea58b2aca0731b01b1f2afd59d7e68301dba60dc3473314ba9c05 5dbd72023a93c3e5bc64772ba0ab1f5f4b5d9bb107abd74b53c01c0a10aeb65d 87dccb6c3ca8208a72379773b9b02d59061da1e22de0683f64a7e2f9ab4d5d8a 086bb4d3d5c10fdd5ce96b562cc01778064497947c6aa99da551b8f38f08c5ab 87d292eb9ecbada0824ae327a15f7bf9d5e29f2308178b588156579888ff9823 f5643f7747b5a35c488691c4e60032f31e58af5909ef3c34bda1aba552d1d6fd 02683b5821554b34822dca1a46f5dfc68ded94c6f69e7da11ba6bdc9ffb3eabc b4e6bd27213f8dd937f20d5f278dcc22d0b887614b2502b3e131f9acd38ddb97 361ffc35bbc8559e4d62ac71fe6314955b602a7c6b62a321ecaa0b8f0477ddf8 00a4b63ac845c674014148c2b6004fa629ff21c3c4cdb40fb87eb9c9a1cbafb0 c6e1cafe3ddb18acd843f589701da403b787cd383421880151c6f844ae92c7d5 b24c64dafaaab3d177e6318fbd63b8f4655eab7a2c78081214b807a05350ead7 85cae579b423fd5229d57421293aa761accb4c60ef9ac799dd022c865b547928 d8d42baf62e432249bb51d4349c11a4215bfa0c3ee060e6ee80e826e1b7dbe3e 3db427baa5d34ad2bd315046b52bbbb5b5279d50e54368823e1e27247d5713a1 cab905fd4b527eb024d6f7cecfdceaa41961e6c929edbcc33cd24d4711d1db6c 3857fec2819dbd855fe3b82f60803f1d404065582449ba9872e1b27a2f1fea1f ba5578b5e8f186437a9133f84005b67d9e211c63b99d9fbc52f5b9e93ecda25b 30d25b0465d33ff19c3d83188a1c95a4bf85bcf4febc501a21a5165d0479afc6 3ead90cfeffe2d1ee3b988ba86a2acbf833fb55f7936428d01a669b19d43823a fb36e43893f24fc7d37125b2186007e08fbea1323bf668b4481be9109a3c41f1 9f15be32080f790d6afa5023bfced90eff61aafab5f3152e603a9a8d6cdf7101 179d3681f230841a0347eb5645b9a115ebead03c4861b3efceefe3cbcc4f8ccf e08ddebba445f6b5e9a0d0b69df4589503ef02a9c7de37f6dc349650c3cf7b30 2487b03ac1c3453dbb69037e25671c9074b48b75d34573724ab57a413d8f21ec f80e0fd5e4b1809e60af7db7574bab3f2ded5f36b1396b6995e12a59a219cca5 3906913c0df3d413f672d14c964f7cbdfdbf7bdac96052ddb108ca399d8f318b 54e5031c0cf6b325fd70bfbf66263dcdbfcd1782f51d72ade178f2e1a2ff25c1 9f2c64936fc82c0ec9ae9b904190a2dcb0dcc4d01fd38741f5b06f64d785c321 d7d5161b178efa11cf9f0a33fefac8346c390addcb50dbd3ddef0f7900e943ea ee8d130ab77d5fe58d5f50b22f92780ba35fa56a11330f3ce23f605c954a4657 30a5f7101a0d7ed08176a2a90d2e8a9e3338ba0cc8b621fe7abaa03891478b24 d09bb4dd393616ad744f538b8552068072f5b0167cd05476418b45abf4ac4ca8 4212a5631135adb2a08b2cf0d1912ff14a5d2499cc1da790941f6f35980a1942 d06ba87f6117fde84fcad78c2ce97de5b4192aab06541a047224799672a6fafd 5cfa00cd942e938108c1feada71438199dd2c05ceadacaf5dc69c283c288ac73 e643d8b7aefb833cbf43cd3422ed3dab6dad43fbd62add8c105a39088962e582 ded703f810323c78ca989e8815173be7c1c3fd48fc03114c8542bac766859577 c7ca9a5ac727c3ddcdb1481d4816ed294d4d8d089a7c2d6a27102044fec68e9e 4e8f8ddad25c9a70315ac91b7960466a0817d9f0496fa575d58ce36e52ccbf38 605b444a409cf68995bbfb3f6f0b0eeb6a645ff057871c9c819e3297c4be7b17 de1c28e1fe88a1173d423b8eb37a1d37ceb5975467534b2484642e77c9f80b03 0176bd697f0480dfe954483aba1a50bbf658bf 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 3 /asteriskmath put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueXX 5000816 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f059084178f5ab59e11b6a18979f258b8c6ed3 ccafbc21aca420c9c83eea371adc20e038b4d7b8ac303004b0aa205f04135140 76407216032fdd22e6219da8f16b28ca12524deb7bca073cc5eba65c102a5e85 fd48e6d062cd4283ee570a7774597e5bf0e3400b6be72db0115f3cb12db70ce0 83722870cddfadee715f10f1fcaf20e06f3c54afe5ca238539bfe2b596116e83 f5371ff18fa5003d8543226cfd4025f9940365b392a858d27f078d3abcffe4a1 54e78c7692d1a32bf935967c64f01b24788ff8325d61145e2d4a489fd986fb77 38e6b254522c77ca2797a504a9ce4676a77ebacb026eca94dde5922c936f8e90 c43e2851973f31a3280c08220536dd2c2de1ffd15fc739091fb7cd5fb9bba2be a5ab0785843300a48f080377b45809107ea6c4e3b0bf4e928ef65d468a19dc1d 34a8be67669f93aa7ca73e2eda716c26be7f59bd96bede57fab460fb9288c1fa 65e8a9b4c88ac5e57507c95331aacbfe1f62fba87b107c820043108dcccfe37e 17de40413b4fac769ccca6810fe903e08efed51303c3db661045b4d6aee2089b 422251d6f63c3ad004504aac463be1e4ea5cf2b048263022c093e14d7f7dab44 3a3da51b53f5e6bced4ea82fa22dd3633e9e6069d0694613a54583bdcc2b1a21 e897954c25820fe0a7f7faf279136b6924b058da2386836ae8ebfeeb572031ef 7a6bf1d70626646d1551d010f8d4f5b731dc8af9419b97c43470cb339f8acc51 943fb46a95a2793d35214a2f8c5d368929ed285887b8d5123d4327d780681c7d becd2fa817f17841fd8ed983eae1eff9b938ca438dd417c26bcf382baf2d779d 0033268eca572929ae82d14143d4a7542be302cc6ab1a7780a66509fb8e2d8da bb7d71d73f972c84a0b6f1e9db85690b8e865f89baf6d5babb6c0dd5127a1cf9 f129176458c7bf21bd294f3c16391d6ccfe3c24a48998f 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: LASY10 %!PS-AdobeFont-1.1: LASY10 1.00 %%CreationDate: 1992 Oct 23 20:19:17 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (LASY10) readonly def /FamilyName (LaTeX) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /LASY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 50 /a50 put readonly def /FontBBox{-19 -192 944 683}readonly def /UniqueXX 5011949 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930e72218b3075925ce1192e3f6620e5383410 9c674a944e5ff85290c767088ede1e6835bf2aaf7a0e947193d98442808fc3dd 148069ab44a71076493a05cbf836aeb3dd5c78829d1482b3c40c2f4cf5fdd0be c3c3c1e08cd5d21db9363551e5ea23a95a928a532e09d2b59879067059a5974e 4912c690ee71858bbf009a75913237a9ab9eb9d8d683ea3fe066fb2ecd3df7b5 c7cb950bdefac7931a3501dd1dd59c10d45f8f91ff07c7c41361e1fafb605c38 0431bdf3b1e44d21d351cf80d8dc6fbb5131fb8ad1a0e9a9a5fdea6432ec1d59 bee32029965fb5c141768d6f2c624b2dae5c9ede58cb4f522099a96749094f15 07e9a6b977ebc86c877b8c4a0758b69dca966be63e606c8efde1f65b295997e5 1205bef471d087f51dc9ff816ef9f6a4a23226f9305feba8da8baa909e515d85 b60901cd8e958e0c94d2a64c1d9510e66d37d18fdf08c5408032c63ae7a667bd 1ca377a9eab6b0bfd434ccc71b779a6cbab2183016bddbc05f171719794bf19d 0118b9e708296d876e7ebd011b0d49e51b56e10bae4b5f2d07b16e0c83df55ab 42cf85da87350d752cd4522682b02f82c49d08a2e0d47d58374443a63831e432 c48f945cc5e42d807ba86cf3e7b3b09f636c17360c5c9bd7717f7e4909e8bb4a 7efb7a580cc9357613467b64fd0161dc026602cc382bdc1779ce8ebc0587f172 654aff1ed908166f3755ccdc64a183e50dc8ff8b9d8ea9ab556615ce390b4660 1749955a3ac5f7c061d6b9948677ca68373ee7c6738abdd0856c90a91d904c91 914f3ba09a3a8b54fa319b1ba9bbd7441b69f0431716a4156816329b7e09b57f 495591d0d9c9049cd0475c7d2b75a8105b99f34816c75f82922b2ff19f835843 adf237dbe7d8fe6c10f1a7886b55baabf55cb33784653da0ac3b49692b348e26 0e2995ae562bf8fce2a4754886dfaac090986bb50eeff294d171eff1b12356d2 e67a7040f2ac7eaa0fde3cce8808b041589918d213923d556f42dbf5c27a062b 4885c544d36fb68419aa660bc55824a647ee56d3a4f4c48ee9b6168c721b54fc e8c202833282c38447a03eebad4a0098fabff640b27298f9e63dbca777a37849 d70d0ae5436463a8d8272bfbab7dee85243e4046f8af3fb02dfbf7fc9fd59837 9b3af8cb2faf3a58d7aed120e3c05f5fc69074b299057f821eab 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR6 %!PS-AdobeFont-1.1: CMR6 1.0 %%CreationDate: 1991 Aug 20 16:39:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 10 /Omega put dup 49 /one put dup 50 /two put dup 101 /e put dup 102 /f put dup 105 /i put dup 109 /m put dup 114 /r put dup 115 /s put dup 126 /tilde put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueXX 5000789 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486858b2a4b5a0cc2e1bf4e2a4a0e748483c3bcf 5de47cc5260a3a967cac70a7a35b88b54315191d0423b4065c7a432987938c6b edad3b72ad63c2918b6e5a2017457e0d4ebc204b031f3fc6c13d7da7277a94ba 018e9998b3dd888011a5d7c4204989f30f908b95533bda845746b673ab71ea57 65a0d14f4350707e47c8276305b28513cbe1bb0dbd269a53719bda46e536685d df78ca0146b6b93e760256b74d939d4e35b5e77238f04c92298dfdd188feea30 e053eefbcbb52f2011772b3aae39f5805597bbc1e8bb75a446ce014030f4f2f0 f49f9e962ee4a1024a746fa92a3628db5270732b54e43fe5ecfa524f127e5fcc 788e77e66098336ad67fe4cccaf0253272d5df79864bf4b734cb9a5859d557d8 bc11b8e00221ebc12e97de4b1f466ead83a4c894709363bca9040410a52d592e 34ee40cc7e5efa920546b981aa659513a24b1b85c221a1875b62d0b89e57a368 321b8043a5b094e0379760a443d632892b14ad6d19dacc8c78093243ad67e6a3 08e56e6b68412ee690b10dac6e17708754a00d51fc957b500eb80175716eef4b 2ca1ef867614659bee3f2b7319e97b6fdf1efc847bf3cee3156f72f21751da8e 5fb6898919e6799820d3de0642d756e09d6fae4ff08dd3deda3173bff4bb11f7 9109c97ddc05897af709ea199a90fcee8ce4c7a3c15b18170c41c04de2d3fba8 f34296a95b8e1e8de3739b17273f8f2c85e914615e8eac5e8bd2387ba3b1edf4 7968f06e2067d836d0f9f3e085cdfd2de06a62c81d786b304326f7002e83160a 36598589228b4dddddc43c85e1d126f8fe81b828028e26317af5894aaccf4f69 6301e1a9fc45935d8a414957f08febebbc3a72ada80f101e47447d019ade56e9 f4fab969bba2b44e47399fedf5caa1bcea216d7ba713d523dc2b1a89801ad4ff 2a26c7929cd309eeca94b69eff834e1c150dfe87ada942fe0f1a05dcaf9ff381 08f9b95b2f4f2036529749026880e261ecaefe625b9a98dee7faad09c5487108 53e7a5cad0eca42dd8bad2c2959868eea171bb9fdc79f50f690f1ead3ad400fb b29a5dd9d891d96c8f572f32d60ffcfb03115d59c0ef447115089ad11c8aa55c 42e112a599c76bf8bda6136b540463e00c09b36e060c0c5215be2ce825187c31 99d4b3870fb91b9f4429b05d9454bd28fb5d467fba049583a2881d56a0df262c b3cec3c8a25b7c3a7d79fe5337aa1c08cbbd389ffe182dda26ace210c1781a2b 1acc0760261ab18c474c8b1f02a7a30cd880a31791583c463dc32c9a691c06b6 7d867b63cc490a4a8cf8a67f6c89f92e15c11128a09136431941031421e18971 c0b76da1b295b2675f27b2591181b3ad262e11fd5c6a8625babf5602cf913018 9670dd426b6a78b0f1bc92d2a4a3aceb3c7cce1d554f26a2a483e4ae93eeab81 97a2054b6e5e58b5a4344d4b6e25aa077a2203881ee3e2de13993e532ee9608e 548dc3e1da13dec51a95c55ddbb08f8b874197b66fc11f7bb4b2ae56e346a1a0 60e67a8f8e575ffda200a791c75ea685cbe96473b60a56e1e5d0f9140ab8e2e4 4d325f97651b08b17404cac76af5e193c11477a58ad0b5e65b68139daa0db95c 45eaf4ddbb6b039b2b11bd3a6f7f058c0099183b716857fe6d033671883a437e 7126113ed64620d038c9df974a94bc06f2127254cbe952f1c76d0de5c1a778f1 2dd593478bdb121d40ef5b659449fbd8fafe09b50de40d70eb001fe8f83949b2 2d7b91a1b565479f41759ae79fc7e13c9d5be761023d96b2c01155bfc47dfaa6 540f28e7ba94787a63a858a5eb6bbfc79d47be991984d64702de04190d5eeb8d d92ae6372ab79996219660259f6c3047583b6d456e9a9fd9092b2dd1b5328a4b 4df38c1236d2d8e36b4050bd715d62142275f58c5b7baca8fbed50691180c73f d573c4aeb1ed372eb9127e54bc2c5de40a70b0d1d4ed81dd8648e428d8fd4d35 3857b08d02143a055a82130b62c1db4d881b34c797c7dd8432f4d73ca50d40e7 eb6b961cb1e9f77b607798d4e56a0219158a5bcb408a4fb7191896934a279f20 299a674f55dbe5a84036d6df576873d3b8262f6c591bd77f21b474d756e6b44d a5b628c32a67714313bac5fce34d94a1dadcb8b8136a5f32c5f5e1c3206bda64 518d9d0db11a33b7caea58a87f6120c63973a02356db49fa503f046eec203b81 85198b7bfc252a89bafaba20589071d5ab7aaec38583aa697d97ec247d34e1e9 daea68ccb416f4378d8277d3b06609d9adb7b642d752482d2f78b9cc2aefbbe3 0df24e36c81d83d36e0c25027973350a55c938c2743ab23791b17f8e8f6816d6 ac62bac33a001c53ee71db6493bae1ee4accefc4347887672934dac346df397c 6c1726f0765037a9e85e38ed08003e9bcdffa48285361edad9c8b428e6be9683 0e1582c68a4ae00e2f4eea0b577421cd8647174d1dacf084ea856c62bc21b582 311a0490e8c70f0e7aa2bc4a9279420f0110dd02dfd5a6d27cb1a65867f2e1fe 67000e080faaff9360fa0135a532ad1acdc683dbb0db174e8117e358563e9522 743faf5b0a4acf5657324b67a614a52ec5959d03664d239852c4223bda0ea3f1 d8d3ce91652e567e4b72c9b53968f6e6516558b6db43216843babdf48f4c1766 92c6e7b4d79fe7c2a583cd2ef1275dda4d95bf41b3326e786427f4b5a6843c55 81f05dfee0dd3a19d354b04bca9d7aed2c91a2b3766bf433bec5817d7fdac509 0c9341b84e1d6456f5c7bd8b306a243ff0545fa97de4be61cdb68c709f7d8300 0e9be98d404a79d3316364752862dd99238366e0b239862f6c0fb330d97b88cf ffba7dbbd5482729fb449cc70091b38645c3d4f42d12562f29633cc56641ae48 da540dfbe633b64bceb858de1ca31130fc3b27301e6ae74b03f861b929d69d5e 956919bc5424b0a38db88e9419d5857c07a4480c7238a1e8ab45b2fc43a2b454 f6d5472d8f71e6c3970c8fe9efd3c3a00b083fb5aff8b1b938a21f9e895e210d 346fd86ee5bdd53cf20002bcfb339cad17f3961f9c05b9787d66f9f489e2cec9 a1fb4c17ac9837696e6f0840e53041bf8fd2cea8a2da8f5cae6024b8946d3b4f 5198fd2b4183c0615caa3a42c07dc2a450e4ba4f7f195f3e2b010c3980d7ef71 f8d4246a066b1dffb79aec4a8b6917811043d416c889d944c4c53018cd206727 06ddcb254f7972f4c157b20175780ad6a4f383033797ced16c1f4bafbd07d47d 5c4bc84c03095564f4d63ca55e4b2e4ca1dd55f07a812e4d1208cf273aff84cd acd1ff2f77ff 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI12 %!PS-AdobeFont-1.1: CMTI12 1.0 %%CreationDate: 1991 Aug 18 21:06:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi put dup 14 /ffi put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 58 /colon put dup 59 /semicolon put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-36 -251 1103 750}readonly def /UniqueXX 5000829 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacee583a5c939393e012707b47e0c1f a47d284a1edc9d01a497d772bca8c543388e6dc0d1e2c4944740470e0914f65e fb0737b0851b2ba713a9a00b36d07da6bcb52920b9b59efe587734027a3c5e65 66aad332fe6fbcced1417802822a3b81d6187875263d6bbda04bbcf6e4870fee ad60f104bb3c5a766610dd11aea64a6b107b0b04439fa2888b8cc39232bb83f7 695aba81f0260cd5248b9e649cd803271dc8bb1656323089e9e2bb50f2b95088 87d31dc36e555668d9578b338402f8c259f6813b0b71e6cbd95e19c20ced8495 2559e47577e612a4d94de3c1b4c6c41a6a3fa9e2313457242de74c7e4c0d722b 3e224d18baa726369542682e9459b951b2ac638494880c42be4e2175a3c067e5 945bfc1a05d3de8c3e8ab97f80ab6000e1f78fc258b85d3c84b038d7d0f5be85 0d43fe03e26cdfbb41c36d14cb552e8447ca678efea1d275e85503123a2263b1 4077587683de69a8f5a88d666172668411d0d0fed3e037ee468627207505dbb3 59cfdc5d7f526fca5416d741073284f5dd2471e09f5916f4bff1a90eea4d1283 7731f6a36e5ea695f4efc3bf5669b7f60ab606df5819149c0c8a4817f06dc172 0543467645daeffe4457293e0c0407c4ec7857558d0452de2ac773f141783edf 81b30215ae8d2d062aeb3cb69eb0d62bf04780997c9c027a22896f456b784378 7f2ccdd4804375321e7946d3d254521249a4d28a360f4d3163912a688c16e807 15d0440bd27bb78fcbc89bc28028aab2f594f8b641e45b9c599180b69663eef0 03465a2425aa4f8b29dade493002fb335b23eb145108ef3ab828755d3753ea8d f0683265497b1cb5cb484c13f7a4e003fa6ab3ed5e24fcc6125cee4e33936122 3cc907af3c64e5aaad882c356e673e6ca2509a71920e99549d8797e9aea189f7 fe18bbf718017cd2e32b4652acf12f16e785462980f5895afc917ea83e9c3eab 6297f1de23785f79172538b39984ab261f14ee296a42cca3159e63fb30583af1 911646257a99b4162635fb79e25404242cccc3bf224d4c279e731333107ab2cb 2598dc582dd185669c1fad35d2be0428389a4b175db159508c08a77e842fd400 b9ddf3d6b5d52e3395118b9ade6666029d48db90b85d60de41e523acff1e87b3 02755622536a34b9e7e298447e979675616626d6772b88a5baf532321fe2f460 bae203dc9dbde35ee3c1db77f9e77acb91e07e6e5faa6428850e62562fe240b1 e6598218141121f9080d375f6c2b69a89a08ef0b86baa7c42fb352269f6f961e 305e2a1049a9795b2c5ea31cddfe2b50a6acb20d247fb86d94ec9eac3f5193f2 a7e28dfe896c0c04e03ac9892fad56dec2456ca472d2afc53fe10b81b08d11f3 8e14199603cfa8ef3f90a8c89dbf16b803abd5cb0491876b420ff91f34dd6cfb 475ffaa47d4c97f7168223cb33a84fb2cc316480c7a69a8df7f1853d295e9d34 29a635640fc14cb0797eeab9027707e7659bee1624c659c7d051f594dcb43830 b963216c3ab624c1f8a5fb6ad79c5ba9e6f56b7d1fe7b01a7b651480da9731c3 faf772a244b00b5bb14452053b36a40881c78ca5d7c5066f7e815c19ed5049fa 9db09d6b952aab2394facfc589974c321687a1cdada42b27a2aa140625716802 2b0a5d3d6c49236154f8168bb012038c183b836625e867df355bc1f01cb0146d 942787b846115fedc6000ffabae8809ad4de22d7b377f2ba8f6fe0f232bea397 4b438e726d30351fda23666f03155c94814a5c5250e53c5cdd2f3a3a836eb0f0 311f3baf4412d967c8f3888402043f96ccb8054c3b4e23f1a018e2e26326e104 712c1f298337382a5e6073c3902c48919d7604d61aee99d42852b07f07d60072 7abc5db730e4e9c36ee4ac55951368ca1a64d2407df299d5725b8ed6f2aeb86f 84fd665470dd42582378401643307e7b0ddcbfa83b674b2479613798f6dffd2f b645bdfb477d3deb192433bf7dd81cf7c91c94ae39618fb3956bdb9a446142d3 b175629b69aaca7374ebcf3d85d8859343408ccbc89f67038bd6a376aebc96b1 9f8b6c57976ba3fc058709d17893a9e740594c23ee2ee5c9a02204a15fac8578 f11389f663ab1a559c7659ae2ec87b8e894272944cf6fec2b004b5f3e00605ea bb87314fcacb02596418ae767fa5a11af8687c63a2ef344a57d934dabd788829 a2d4871da7a92d739a5beb7052eb9fe988cba040b2ad7449f4d3d80b01a0c1d5 f38eff500995975a95b8de89c12ace86e745898592dcab8998ee8fa33ed6c5a1 f143d5e5ac30f03fce3de00ef5ad1b6e776a9d2a378e5bb45ad1a6e9e4157868 5e4dea08c406f37043c21a7ee7be300025d28e48b0e2900889f40216335e23ec 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e2b2cba686cca980305c210670c982106af58badb0742dc04c2b7ce09f2d7731 92488ebcfdb824fad5a7d319c932e8dce011c1260295c7ee568a7bc3d23cdd10 8d4e569a6d71c8b0ce6b1a9eb6b2d7cede82acc7df3d3344cae4784ad6142f88 fe8192c58e1c8b6ddbbbd8e62acc6ba869c09a8571209778a9bb01aea34b8079 dd366236464eeb2abe17c76e1d5e234ba8c2161cb8818e12ef5d6fda6b8d8843 b315a17f7bd100c0d45d0ea85b375c69e1e4d792ce78d06833d6575174f55705 9b85859eef6a8bbf7327a463d57b2993fb2d7a0a761d3a2ad67d80a9230fc62c 2f9cc6303b254f264643454f7a8fb3bd3dcd67a052c58d49daf9384c78f9b8aa a941f12ee998f56488030769bd7f87b1c35313c480908d57c6aa8a75eb21f53c 366a0e9b3299dd9e09c33afd817ed1c258384c64126525af0f6fb92eaf1a1f14 12c74508f1439f39b5183a6f1a866a89b66fbd0a496b0fa63bab40b7cad547cd c241649e88e61a14468e0b511c42493e9ffd16ca907a54f13f3fb9334ea2bcbb aa9deca92f035bb23760ff0da7725bb6e7b15787a6d132da7f31b68bcad55771 60d3d225cc5474aac285702d96826897d22fedb41c3e0fa33ef817712c7c29fa 67aad3668cc4910619a419292aeecbc8083548dabf680426776b7a14ad675ff3 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0db159d8813616cb317b7537ffd57f942c4bdff2270e41c95122e354eabd3a19 2eba5d39320004118a68ab1394b27a4126312d6e88d5683fe6b4 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: EUFM7 %!PS-AdobeFont-1.1: EUFM7 2.1 %%CreationDate: 1992 Nov 20 17:36:25 % Euler fonts were designed by Hermann Zapf. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (EUFM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /EUFM7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 77 /M put dup 78 /N put readonly def /FontBBox{0 -250 1193 750}readonly def /UniqueXX 5031992 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6f33457e320fe93acc64a4016412366d1ad2 8776cfdce37d3a89b95e5bb9373a99730cba9c11192589e22438b44f4ecc6a8d 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI12 %!PS-AdobeFont-1.1: CMMI12 1.100 %%CreationDate: 1996 Jul 27 08:57:55 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 15 /epsilon1 put dup 21 /lambda put dup 22 /mu put dup 24 /xi put dup 25 /pi put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 30 /phi put dup 32 /psi put dup 33 /omega put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 80 /P put dup 81 /Q put dup 83 /S put dup 84 /T put dup 85 /U put dup 87 /W put dup 90 /Z put dup 97 /a put dup 99 /c put dup 101 /e put dup 102 /f put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 114 /r put dup 115 /s put dup 116 /t put dup 118 /v put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-30 -250 1026 750}readonly def /UniqueXX 5087386 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930e72218b3075925ce1192f11fc8530fcd5e3 038e3a6a6db2dcfbae3b4653e7e02730314e02b54a1e296d2bef8a79411d9225 dad7b4e6d6f9cf0688b69ba21193bf1495807e7bcb09b7064e91fa0ded228e42 09aae407a7aaca60b1076299ac4abd23ef02f108765f0e3d91f92f3afbfded37 2fcf6e4b1416901517da8f2fb3c9fe7a87bcbe6fd36cd5b5823fdb74229036a6 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 14 /ffi put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 70 /F put dup 71 /G put dup 73 /I put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 124 /emdash put readonly def /FontBBox{-53 -251 1139 750}readonly def /UniqueXX 5000769 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486be79011d1f5bfae5c1f476ee6f05eb1d2caeb 269958b194521197b312fcced4867f3c8fbd030bd715d8ffda1dcd454b174e7a 1a97b59fe770e67702519d9d9b23d61ac08424d555242a8ca08c49aef300945d 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 61 /equal put dup 65 /A put dup 82 /R put dup 87 /W put dup 91 /bracketleft put dup 93 /bracketright put dup 99 /c put dup 101 /e put dup 102 /f put dup 103 /g put dup 105 /i put dup 108 /l put dup 109 /m put dup 111 /o put dup 114 /r put dup 115 /s put dup 122 /z put dup 126 /tilde put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueXX 5000791 def currentdict end currentfile eexec 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ebfd9e6e16b414cd4c3a1e64c32a96f01e6ea59929b1bc3157d35ea168151793 e6aa6a27c5b81c6b9c5737e285f68c65346ffa6627ec1fce4ca1c4683744e740 660ec7e731ea5732853a21b4c01799a87a557a8b0684469f2700de8b9e661740 3ce42b871b8ecc5b0b431e1fc73734e1e727f1278e008787917ad25ba3ec0f49 4a4297531a761e6f992b82b1ffb3a77157342bd02dcb143b14145e4460689386 84b6cc432481a24546d581be601aa8 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 14 /delta put dup 15 /epsilon1 put dup 17 /eta put dup 21 /lambda put dup 22 /mu put dup 24 /xi put dup 26 /rho put dup 28 /tau put dup 30 /phi put dup 32 /psi put dup 33 /omega put dup 58 /period put dup 59 /comma put dup 61 /slash put dup 62 /greater put dup 65 /A put dup 66 /B put dup 72 /H put dup 74 /J put dup 75 /K put dup 76 /L put dup 81 /Q put dup 83 /S put dup 84 /T put dup 90 /Z put dup 100 /d put dup 105 /i put dup 107 /k put dup 109 /m put dup 110 /n put dup 114 /r put dup 115 /s put dup 116 /t put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-24 -250 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1fa638c0ae75a790c448a51553d2dbaadbb307192383a21b6e2acb7d93be6a00 a6feb5ede3886b5602c25f7847cfe7a59dff251984b2da5f1e2d0bd5d522c58c 2552b136b414251b4cac2d7f1f7b7a3aa3b08c5c7bdcd0171a055fdc1ec04db2 a46ef2eadb7a73ca89b4e33e3a06f8e11cb9f504081ba26932ad609c35c228db 92bc0949667b2fc4770100949bc0f6b263f328dc0595dd16833fe26a3a81494f 7d2a40447d1082f0871838318aaa7cef41 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 67 /C put dup 78 /N put dup 82 /R put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueXX 5031982 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6d56491bdecdcafdc988206c5a457a19270b 37d0ab776e03eaa7eb568eeab6b5e79dec03b0dcbf923a2aa8e4f4deda2cb043 858d8b430efef0ed914a80ffc9818aba0fb30dae3694e5b31df3855892d59b89 7f82ff79fce8444c7926ffc4ea008e63bc518936c9098afc2d1c14030a2c15bc c9285c1b57e80570d2f6301009e63d9ddeba44e3251c75ca9616a2c5de9ade66 1a15db37b40b67c1bd71f77343912c62f47dd24be0e03dd833ea60d2eb020407 eaf042fed30f0c2f01956c83d46d982b50dd979c2026ed5f71979dd98c9c1eee 25054000a5eed5ce9498b2cb59a7ee55e0538ab47d86e69b2ac79229c8c6bfc9 3a24e822a2f1c9964fe630344579a25f66646113f73d1315c447f070c575de76 266c6773b01137e4a039df5e4c2f199e8e3396ef9fd001eebe090e04384c4089 ec9f13a07ad8733c6af775f432f154ef50ec4477e08760c03c006e4d1eb1c7cd acb8955ca4b0d62c7de046ff58d431bb9883d28586552c51809e3ced4dfc2198 b97ad9e52ea33c91b9e335a80d9b77d3430e0487930f992459664abca5dd9177 150b0fb502fbb04ccaa94ffe2697ed07906f95fd98e7b325bfbf10ea6b39fd76 c1e5295022cb39e9cd471b7b0b3f9d72da99deb0615d35cc22f42dbf5cfb70dc 6279fd520d99f32201abf25e621cfe1d37961d9c0ad36ef3d9264c547638152a be30812b1188a10f6327c9e8714449de2ae636420f56a44cebd774f0b131c629 818455cc8dd8d40939f3ba8aa5abc54590c7286b172193c23adb1b4f9931a9f7 73b960b1f5b760438028478e4442dc93a567b07e20017c18fca2a1daa2649dee 5a6ec51fd6c3f4b27367c31e95a56ba4bbeda0493bc02306037e2ba69add9b94 373364db316691e73d6811cd76fbc92b0c67ca18dcdf05807f17d1b309456ea9 d113ea0fccd6eb73bd8dfdbd5905a5e38bbf61cfe58fb01ec95a4bdd9565433d 4a7abb8dca6f76c459484ad29c3480b588fae3d782340da9ccf5a3d15ebc4b0e 37e0e981f80deda1e9edc94ebc4269488bc7d09c3feaf61f454abe05a5c41289 f919b23cbd7cc570de9de9854d6be3f91530fd17c83115c02f6b34beeff7942c fae26f720a6defb2716a35ceb107fb9fdb87b6bcc3481ae05e08ca8eadc508b0 4b7ff96b6d32f54dec7e8b464057722370f5903fbc264c89eb31ed7e18c1a047 a7c173d8e80fe2fe10bfd4ef52781a5ac7d618463633c73d8e414e9129e2f05e abf6f3f2e9ff1a5e48863c810ba563fc8c1a1f1945422414ba0eb4086eaf68bf f0eabc6258f2d9015cef22a54c9417a293521eaa35917ff95937ae1b1a0dd3f9 98835ce49a704000529dd9cb93afebbebb69155c0b158bc0aadb786f1cde5f35 26cb4dc0baefde30ea5e277dc17e6037df3fe03283cc889a55498040985817ef 6af3b4d335e85314aa81a6fab86184ecb0f9cd53ab6b52673c6e0ce1afed5e83 50f55d705184b3f2e1f1ef51bb8b217e9f5c34314b677b375e7099d51e0b79eb ad84d113b155df64e6aaa5e1a80427862cc1c4c7ae5b6bd3dffa0f5312c6fc5a 66c6015faf233d08e090f7bd1907844c8a55b0d078afe863ab50469b5101ee45 6132a08a5901e40e3834f9c0bdafd9c29884f077783a3c1a1e9a6742f73f0d1b 599f84733fda79fbcc7a2ab64f3e722b6ab3a5d085d19b53bd3663ed7f80b2ae 650101063e7033b1c5c29ffa251605d6f4f7bb4a086068823a7bf37b091002e9 6ffb935b164d92835ebe08648d7312a5b26b82b9fcef182eb69fbdc7bcfbc509 a441d6e9f815949503bad5deee7a641293d740e6c942a4f4477f16fb6cd73296 e755c5db4b7246db87cfef1cbcfc76da11e1ffc136eec32a4bbae4e17e88aa30 b5b14695fc99a342d3bac179ff423edeea8dd6ae38c20120922daa7fc9ef1a4a 372067795107cf12b8ae6297a37fa4ac7056a97fb38b6f8ffe8212fb8399dca1 f7b6c29e821a92905579eae68c8c9343a7f5c1c9233f26d15ea56b3be0e9c6ee 717053a5eb4508632e8f93eba56eb5548999b000e17c300f1e7b589bd69398bd 6e0a084ca920a9b4e22a97d976c93edc285eb39638af274b5334ac1ac52a27e7 888b666b2ca668d7a54016260725afc68b239a54a918500667a57e96feac5d05 a293f9fb751d306a36f2e409f028fd267a7229f4188213e80449643c264741e0 194732d842ad35cfd947de0a43ef9fb9d9577ffffdbae496c1078aec6f050067 4a0123525ce76ba027b9e1c62b0690c7e2eba30ba3454f6492 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 3 /asteriskmath put dup 6 /plusminus put dup 8 /circleplus put dup 10 /circlemultiply put dup 14 /openbullet put dup 20 /lessequal put dup 21 /greaterequal put dup 26 /propersubset put dup 27 /propersuperset put dup 33 /arrowright put dup 35 /arrowdown put dup 37 /arrownortheast put dup 40 /arrowdblleft put dup 41 /arrowdblright put dup 44 /arrowdblboth put dup 49 /infinity put dup 50 /element put dup 51 /owner put dup 54 /negationslash put dup 55 /mapsto put dup 63 /perpendicular put dup 66 /B put dup 67 /C put dup 68 /D put dup 72 /H put dup 75 /K put dup 85 /U put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /union put dup 92 /intersection put dup 102 /braceleft put dup 103 /braceright put dup 106 /bar put dup 107 /bardbl put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueXX 5000820 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f67052788e6703bb5ee49533870bca1f113ad8 3750d597b842d8d96c423ba1273ddd32f3a54a912a443fcd44f7c3a6fe3956b0 aa1e784aaec6fce08dae0c76da9d0a3eba57b98a6233d9e9f0c3f00fcc6b2c6a 9ba23af389e6dfff4efec3de05d6276c6be417703ce508377f25960ef4ed83b4 9b01b873f3a639ce00f356229b6477a081933fef3bb80e2b9dffa7f75567b1fa 4d739b772f8d674e567534c6c5bbf1cf615372be20b18472f7aa58be8c216dbd df81cc0a86b6d8318ca68fe22c8af13b54d7576fe4ca5a7af9005ea5cc4edb79 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (EUFM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /EUFM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 77 /M put dup 78 /N put dup 90 /Z put readonly def /FontBBox{-26 -224 1055 741}readonly def /UniqueXX 5031986 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6f33457e320fe93acc64a4016412366d1ad2 8776cfdce37d3a89b95e5bb9373a99730cba9c11192589e22438b44f4ecc6a8d 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR12 %!PS-AdobeFont-1.1: CMR12 1.0 %%CreationDate: 1991 Aug 20 16:38:05 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 15 /ffl put dup 19 /acute put dup 20 /caron put dup 34 /quotedblright put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 93 /bracketright put dup 95 /dotaccent put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 124 /emdash put dup 126 /tilde put dup 127 /dieresis put readonly def /FontBBox{-34 -251 988 750}readonly def /UniqueXX 5000794 def currentdict end currentfile eexec 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 3 /asteriskmath put dup 6 /plusminus put dup 20 /lessequal put dup 21 /greaterequal put dup 33 /arrowright put dup 35 /arrowdown put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 63 /perpendicular put dup 75 /K put dup 106 /bar put dup 107 /bardbl put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueXX 5000818 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a 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b416244787babdef53f43cecafdb7170da9d3ea538d631ffcce9976912563bdd 993669e341adafc6ee9f7341d03b501c2ce38ad79460e9158137385b710d7ac9 18664105e219e446d649807e8403eb2f054e0762c8932d8a0ba643ba37db626f a06ae2fac1ca31c3b55bef9dfb607e090150e88e40af2554de9cf3e0adbae56d 083bf81333e6af997bfb8f60d73dfcfb07f5892ce8e135226e062c3b5f6d5237 8d42a304294bd173eb9660c890eeca04d0fc950d1495f1be03ee26f2f0e571b5 0f3312ffd822f3cfbb28f56b83ec710b6cc97b3d559ac50f6ec66e67ba98f487 aa7b3aeab5191000b3dd4314ca019844306e9c10f20e18fb0c8ceb545695e6f5 e6cc8598a9d0e5fdc3443c9298ef110c926b5ea65b27ed6594970f177d2a1544 ae74520263db2fb508927b039c2ca0254853ee2299ccacfc34bfa643480567ca e4af7a0add9641a3fc7080ad4749ca30ae7e815047cc05dbeef85b969bb3aee6 991c5907d673a86306fc0b478ee8573bf9a15b1c17c1ddd7e03f05227c6bf7ab 76502b3dd26ce409c78bf9cb4d6ac91d38eadebf4b06bca055afc5d535a211a2 fefe2e433074f727b60f0cf0e3034d0ac17bbc9222fbd68b2aa0931582ec269f 0cb60925ad1d193d3c8b1818d1bd041f7b92febefc4b5c0584826faedbcaf73f 920aa7f13869f7fc76bc1b311922435b95acace2825e45bcf77bdadb5c717ff6 8fd043bed255daa4020fe8b2b91b2673a577d523b37fca4f61a149530760e0f7 895999ab057e186cd18f0def3ecccaedf70e99a2d909811c29883c9b4ed3f4b0 3ed3131ee40ada0f64b11839ba8687d55801111970f7da0ba206bcfae951ad57 2d502766b4816fb2e8304b7f4538bb0a95ad915f6d588729e339b3e49d0cb724 30af7c60c6cd19e5e5d1e3973e13165b441ce21a10b1f569aff99ab52df1e1eb f65bb4df3b8de52b14a9fa4b0c81b4469518bf940b4714ad359ae4b67a862fb4 28070c50e30c011ef1bbdb2075574a957dc42c7c4d30faeac881fafad27cfc7b f68fdbe173e7eeeec0e4f36943649d95d833976bf25436cc363962cd8140da86 820ff2a77bd57ba0e8f32d31933b78ee8805e27c040caa21b542cfcee5f6c063 5219abef2bbf536c60a5c06aa83ab2581db6cae36bf8187585b24183860e40e5 d0651e101904f967240192c0feb9573889371ba2e4abdf66c9ca6461d2f58859 552edc67ed51cd4148e71b65409f9ee4c03e4f16d9f4aa5e15808100535bc48b 278f88ff3eb467e9426faacbf598102516edb921412d560f713e7d232ba78716 f338ed201a2f7ea91a07f4f693fb042665a88575759d18a4cbe62000a6d8ca39 fd1c38a2d74f0e8d4aac47aaaf0db7d2a4b593b2061a44e04d88cd2f6c0d90fb 80a2f192dc8c42a9a2862f3d293e232089e7290388bcb9f659008c42669b6156 67bac06c0a694334afc8a6936a546328171e8133013305c06e1989f9c33d78b5 a6cb3adf1bac75480078f8a271d610a1d661357caa1d7a78096ddb45f90df52e 0ead5186b66223068d9a7445f92e313dbcb072586f5f94132090a70b27cb3d56 a4b8c8e0ca8369ff5501b3779d9cb1cab6e05db56c078927b59d8c8eea00094e 1113351d7a2055a66c168df9b87de3c5ff685ac8d17a2462881a2256d12ac999 261aa30214b465d624710b96fdc1008f98b005fbb73cefee5a389bef3461a856 b5eae4dfc9458ce7f43cd7fb06f596c525310e9c2ffdd524a71283f1e4e5137f bc9dbe66ee9a4b820efa0c6228ec1c24dec307ed6e2b3fae974b34c6e19bc654 244da73f74a385ce5f51b2fe16afb7221854038dd436908abc8b668c844e7e53 3624b64b703292fab326feca1d447a8c1b9e80f03c294fc41efb9a48d493df2a d8e8cf79e7aa6d561771e5f4256c13251066bfa7e772039cf997ad8647e40176 b9bd1f1e8ab9f06d50d0ee5c77f58f1fb54291fcbef80fee48072875865d40ff c0f5dc795b0ec19505f4f033b1a7a756ddcff42d029d0db6ae69ce5f6824bcf9 80859c1ed4859d7e02c035b1de75d337934d1317928ac7f3d48e4dac6efc865a 7cd1a4d971671591a575e8bf362ebfab4942a172fa001a37355bf8143cff1a70 5a6cf1e9cc30fd39710fda10add172d4f34c35bdf0a62ed3dca1b2b85ded46c3 589e7ab0fafb72831ff1c6d22286f9f05105ff5cee3da41239777c9af328b0b8 91dc63ccadc18c750f5fc0ab751e5b1b42fb0a1bc071dcece7bb6da0969c0691 60c1849d9c46a065a2618951a8a39c7b26b8fd11f1c42117be90e181996b8b61 ee1d56a2517839f6d933c267e53c4f2d311b2f50139885fcdbb23b5374ec5839 ddf27314f7b4566142f00fea4a2484b45783092856f0c4521da2ca88316cda40 afc4fc76d50afd9003d2ffb54295d6e9e7b34c1ef2b5784df005c4351e8f63c3 902a2aa5f617c93baea73e6961f4e3937fbbe764e8ce83096b011fe87827a750 a011d7f4dd75fa5276c7ef61bc6edebd7584b144a1beab30bbb52fcba41c9cfe 25cfb08dca341fc747f44cbdb1de8a2ee5907c6a173352c613f111e19f6a87cf 5468fe82b7dde035457cc19c27d96f588ea1b2539e60ca1cea3f297445d15d48 3d5c8df91a48a758e9f0cea192927a3b254f5d4e526cfad1ea74758348d03621 ffb106b8236b48b3e6f0fc813058a741afde48f63a7ce530cbd78e4e2b8e64e5 3d496522b487f378a035b915df9b9b5b8687a95239badfa6b07a5c626828b7c5 9b21c42cfffe379559ce7a6c081ae723ceb6c03c202f39252b904a 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /beta put dup 28 /tau put dup 59 /comma put dup 81 /Q put dup 87 /W put readonly def /FontBBox{-32 -250 1048 750}readonly def /UniqueXX 5087385 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930da4f8d58abefd45db119b10eb409dd89792 3c6e705479464a4b310b58348c4b42393988fef4925cf984423aaf65fea9f0e6 4629953bcf50b919d968d99bd185f83112b2759cc411764e9bde677f57c5ee5a c555448d5f81a16259ded1e11bf4119d53e8ab07a802df900d3a2d5ccc1c6876 d29c5e9effb7af3ef83400b0910c07873a8c56fa7b1eb1ba426043b00cc95dbe dc6e136cbbbcb3f67509964f9f281ebf81fe5b018122eaf66c4a838487e82e18 6e006093042848a903efb3469ab6b4049767aadb95c30408dfd58d8a10f4cb22 168decd9f3ee100f07b49aa44c92139b669cc312ba20192454eb2375be6284b0 26659d964b96ae82d4942e758027fcf23c25ed01115af27ce7f20efe2a822bb6 84004f20243a49c9e93301fc21b80815c033c3e2ba58ef53da2157d524b395f2 b37abca13bc6a2f42e824ab7e47106176b0d6db267fbb795ac7425582df2e3dc 55863468a9200742bd7b552c48f8cf58bc21343bd3b95abfa140f33f37c6f3f7 8b0d8a5154eb7c1f62ec598267f13e841a3e64172663935ac8b665d86540d316 ddece329c008049c5e74b27d59022c5515059bc3b89370b1bc6a169c888bb325 e0b74282d6f053a50da4024ed1e433271a32ab8c17d41c632b41cf9f3ecd5fe1 24daf7aea7ce8a63047b245822930d517df7baaafa69d2d17f7d93cebd45416d f3459365434123a76ccf883a4973bab19807e1f0c4bab03a45f7ed69ff2660ff 3963e4def11001eec4238c368a39d874fd30b3c14f4186ec7700fbce22abe468 eb32baf7b0164f8f21a9b5e706b91411faaa44467cb2180ab03de375c1ab93e9 76e11ab92fdd4a7280a6d1a8dc65c4c89d04c8ed6988045a2a3de9a4c7b1efd3 75b3a4105c3d10ef7781c955b819182c6c351e3c271a5e84789550ef09a61d28 477f55ed66adadcf1ba915c824f874089e4952cd96f14e59cd96e8532236573a 61cdc9a2eb56f67c2919258ce2f6f4d143b6de0e2552c21a75172018eb51f34e 3ca31f3e19dbfd9e2475cd2045cda8220809b1887b95c1baff3cb28f77313f06 15240c147ab0777bcc25a70001a622081430e462804e3d31a190b21030d44970 032868d54a4b0ae0b75cc4aed436cec660891faa939854eb24453b35b9f6a40c b4e9e8b6eecd8e3dea75e2f733278b95cf3dcf020a7846d4f964276f43944129 71b79e46a5019b8d531578fca7f282206a39059ab2fe277e687c0a4c496888a2 e1c36f0df5172d4aa5821229d73ff5e49a58f4d729e7a2bf973d0af0e747116f 3a2c066a9df69591b1139f6f58083407490e4e0ac76a94ef054994ad4eae98f1 9b4a56c27ba951caf62ba374b9bb6eba71bfdc125177f336bd0ded58af88c3ae c99aa47adb0f1b76eae9170470e9e4584da6da90c5b66d1cdafdfa52dfa2fdce d7f47f37eeb5feead1048c55ec391374075439e3c0aeb1cd5da6407dcb8d72e4 e3bb757a83ab1def425a332e99ba56dcf8bd06e669818400718f7f83da86c682 9b45f534bff32b5f9a3b8b9e3bf68c442c692a31afa73f01a12d9455f8cdbd32 52468683de02eab10c3d606338689cbeb2042f48afacb18bd4f422ff56c14e23 7d6377bce8bcffb05287d88fd2f0cdd297fc589b3034c86fd6b63de17950d7c6 bd13e1c9268b3caa704c94ad0a2013648c0cc84c03d8180919e0213a9670b6a6 c511db8c0eee707b44a9dd5eebea3be4fea81cf4ee22470ace838dd5e6e456e4 4969ece80b7a1f9f1daea2af1bc496d04d15d42d5dc7119551779fdb2b013ce0 0e88d299468e0f8039963843ac365ad9f1cb72baf47e7238cb418868b21965cc 069de4321922317748ac0bf0e0364b1a75c2cb8544960dc555bbb2220a7a0a57 f5ca95774904ae3018ad488d86ef8d4c488b43d2de9a18d7400fa3b57cc14273 4743095f1756f0c09960c9a3491bbafdab61a974073e7e9fd8d6d35cbe18db0b 4403c8982f544c6a5b0be1b3fbfb6e52346d7382b4bb41a09136ced073643e8d 19fc397b0746eef0d9137d3f776d512b1fc45eea1185bb223ac4282ac5ea149b 64b5513a86b6a4786337072c52b6da64d3e5f8ecf11f22a81a807ec66fac6446 b10f1863f0dd8fa424a68ac547bcabd3ad460c018544ffba09f16f38cead41e1 eaa82cfc4c9ef2e9156cab8e3362816a8a08ab615291692d0449f10d6dc0b2fb de0b68beb2320ec5b5dbe128cc98547bd0add5667e758f1624acba5f68dc4840 5d75f2661c20933095075ead3f796a7bfac42d39915f89eda088fcb757b9fd4b f44e20b01837ee85c194646b916a400f3c06ecc2324ffefa48e01da96d5f6ea8 6933f8d5fa639a012bd046a8b5d32b433ef506e2854beb6bca0f621120e4c891 116fa8cbcaae32c5bf0740de92b0f4118e1e61f6e02231973b4e5a7539d97c96 78c2580f88cb7842f82efb861766940d594cc8d20a86df474470e2dd26e0cb32 1631d27801bf4f93afa654778212fc013a5366b98d 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 65 /A put dup 71 /G put dup 73 /I put dup 75 /K put dup 76 /L put dup 77 /M put dup 83 /S put dup 84 /T put dup 87 /W put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueXX 5000793 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 b9cfb856c23f958f3c5a2fbe0ef8587d1f5774879c324e51fcb22888b74f2415 50d7401eb990d4f3a7af635198422283cac1b6cd446ddbcbd915db9bff88844e 784c6bf7389803d9450b0c21756a017306457c7e62c1d269f306bd3402e266de fc3b5e7d8a8d2f5bf0fe6ddd40d07391df4fad4a6018dce29a2b8f692b29f202 3a7c0e66de8ed85c14f1f8492167357f51a7e84cc5d92e0fee4d81cf7fbc8de5 2d2e7bb57142033993f9c08c315abade8dbc4a732e84e142d3bee51557910e12 cd8aa37c459a5e6b7f5269f59078aba3be4641a11ac48d0b625c8325b38ec08e 4c9e5e7fed976a5650d99d82114f449b9ca14c4ec957702295a39a93ef93f618 99b8ea06b092c3c1e503e6e436e0a9fa22576c8930ab3dc8c20f5d82b69cddf8 ff4dacfa9c54bed5a3aa3ea5b129fe96be632843b9b6bc91b615581a985db56b 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58f007924ee84dd54e0fd96fd651fa29032d6886382e4ffc7b00e4ba4c52dc9e 9e8099eeea3583b219b4b03f3f6f3babceb3d9cf1d4a7b2c244f11ca670df1b1 fa83c8e850d6e05fc71b1a0530ed3272c6312eb84d279936f383a5c06483a78e 821692065847ebc147f04d0131d26366496c9b64f3b6113172f53c6e59fe7480 06512fbfc74059785eecc9bd89537617d863643b41ba2f7fdcfe5e2aae5958df 16799d2a6ec39f8b98e85c4c9dc9f84aa4fb01f2678ef72657808239f6b53226 9e56a95ff2f65cf2a6b69f4cb1dbcf1a71ef29c53baddbc3dd891c7f58689c46 a8856777c5e0ef377e14318b8488617e32485acd455460fd688495acc18cbb25 821a239cb035a8b95b0cc6e18bcb6ea0b7633eb27f4409c363f35c69be17e50d f48e77904c2ce02e430eb6c72fc958be70956dc34c36539e958f074ac22ed219 9be0855fa5443353e30bf443998af6e271080f8119a4b194efdb4cf7186d08a0 0458cb67950c8608b73bbbaff226ae2532c760310a4e582bd1f8933af0135be2 24f9534a00a7c9d958ead9459c323576aa4f4eae5d41ec99b202bf07da86b2cd e019af64f4feaebec728f7487582230045d0037b142c596cd4eb1aeab7002bda c13ff4b5e504bd69ad58a97b46a2b14c0f14cfa56f36de32abe0045e3ce620e1 361ba33f1187e3e6bd2777372abdb15830ae98fc4ca4c034f282b708864b0805 0eee0c2f97c06a5fd09556bb7e5138ab5ba5003c328e04a4091ecc34860b8398 cd9c25c677ebdfccb79c2866f15810c7221dfda2b2c8b6a34a1a0dab1615df19 ad6c7fa52fd60adfc23005db065f2759149a7715a4f436f46f8e392f99283336 fa256f6eac9b1930580347dca1f7402170ff1dc9e7a75f476df0cb95d2990882 8cd1bf38562d192be49cfd7cfa8e687bcd39c32d5433cabab77566fb07a4f68b b3290958413cdb1cc71d4a8f6301a47d6045d00adbf9aa58ef7b04179e1f6a47 9e27a8a35a6fb4e89a1636ee7e4519c1b818844c8adc1be43687316f608d7057 e53d313d7892c919a5ddbb670e28f93e915b1651b84284e5701e4522af0c1409 e2dd48327f0391b840685f94655e09ccc4516dbdbfc644cf52ee5f217f89ca9e c1472d7cb544976df8f03ce1728ae01292a3fddf6f572a32d234e6cb407140b5 06b7622e9095abab01184f512ae9feeabc8134cc35806af2d7d1d57c3c3e5c58 821f1e38f4c959c3f7c5399783289f7b7fdd10b8e7c421e9f288b37b21382daa 491aaaa1c1bc6ba8a5f2d75d64acd5e2bb38cfca957b00a9dbc0fc1b3c0f9811 e1cf7cdd27558423e145a88e52b56e6635592b523854bc5ddb70e35ec81cb9c3 7a3bd5ead6f7fc0061d385501ae8671bf8079c498c93b57c8309e2817f49b2be 376f6754a9487aa83b075868f034de0f9423d66131bfa71bdb60cf748ed8d3de 7838742f3a89ff0da353b531bc10ca88d2e149d0f027b482dc737cb239233c5c 4d2454851f2d85e9a0e008af5647c4067287144ca8ccf5a995d48dd83a2f1324 f23d53fb946f0d798c4237d9bdd00b11c24ecfb2d444a2e0b5fae722f86bdb03 b85f4b64362a40f14d2ee2181504f3e81d9d984d66d8d087201ec3117d417fec 4ff7bf9dcec9c8d4545267aeaf1526a1a25bbe39f491005685d416562427417f e5e4dcf9579b6582997e8d15eef9d871b3be1ac31a32c7a78682c64eb24eded0 779963e26833041eaca1dd5945f0a5a8a446368e18a19aaec81011aaaf4b4ce5 ca9126fd92d249a3 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 65 /A put dup 97 /a put dup 98 /b put dup 99 /c put dup 114 /r put dup 115 /s put dup 116 /t put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueXX 5000768 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 b9cfb856c23f958f3c5a2fbe0ef8587d1f5774879c324e51fcb22888b74f2415 50d7401eb990d4f3a7af635198422283cac1b6cd446ddbcbd915db9bff88844e 784c6bf7389803d9450b0c21756a017306462c563d51ece66fcc9c831843ecae 1fefc1a232e2724f7baee428ae03aadb95c3035345c15e9922fe49e1f2cfc980 237316572dbc57064edac9b0db8913a5e2d45e97e19a91435ccf8adfc835b585 48e74b291446d689c7f2f8c4325e8356e974ae30c3e2977477baeaa33d141fb7 80f59351e84bfc88c87b3a4a1d25e168b9d134554f6a581378c7d2d6eca8ac09 045cc3a5e0ed86f147133f094029e3483bebc81cfde69942cf645ea20d0a2b64 5140ccc2f9cb4a3d969f4b0a79cd3329b440dfcff5285494406f3dd08d804c86 df191e57232f6f1f83df6430f214e03fa5412b1e13d83c60c52a698bc9a38159 19a64162af7b83a9468fae9507c6cb55ea80df8c696dfbc3296c663902d77e89 a181502627281181ac8624f65be9de5fb32f056cc7dd7cbc7366a91717b5892a 34ef6b441871c6993da3a81afc9bf70ff3842a21a29c0e49da1ccba37dd6b920 2702db510b8b1109acf4c27ee8d676bdbd16abe189f34a9b61532eb777457698 cee0561838eaab2d3939c5093afbb215bac433d1918c739b62566906bb2187f3 378521fb52e70c18886e1aa15f53a1f8a169528cf2ddd54c203613ff88e619f2 6c9c50c2c3b8b8c868a88afab1dc8472e7b6f3f992a161723a3c0904865a7c42 3281601eee91f35222c4a71f343739f102eb43145a47e4fac3bd666e4c114a90 5950618ecb1bdc29f90f6ebed1d8325fe6b52e14a8870e9b4bb088e585c71b91 493c12845a101937e3473ac5e9ae24f7c358f06c8b42dee8ef2a1a7a9f536037 57517f2fb6da0ea305d9d20cc74d922ec4b95b8b299459b71825f6f67abee3a6 f5267c6283c7c62aa8d36663071f18e1a3f0b35dcd88d739caf8dbc344630a28 7f583f037768d861f37d7929f248146b5f31ed9f61ef0a9659b50f47ec328516 9fd999b19687e50303557945476fe9845e5c4780045d84d7df253fe61bf69607 fd277a265296b02b4e4b35884f985d29521ee7327887d983bb92dd84ff646661 2b348b5e558d3bc65047afcb202750ea6e9c18a769a4e945e0954787ee65948e 7e8e443a9f2b9f5408e0febf523423b2200166ade223570bd17a6f7dd0386f3d 2246ebb40648548e0a3ff7bab6637a16137c24b042010a3cce2ba9afb9317fcf 4a5b893c4915859ff2b53a77ff0334966f7a234ea8a4349c3e048acd6876f5f3 17b9c238f37ed4231b5e8b661ba52c32821de47f7a0f6fa9c7e0f4bf09bf98a1 8008087aa44eaf37894e83891f552626f1f46563b5188ad949521e1a398eeba3 b9b8f4ac3c24785d50bdf01e5c9b7e4ea1d6b7b6c8019d2d5f353cacf216e30b 71f09d9b95805210964ccbf8f2e30aa04ac3a3a57bde635c9d2be1f5c8f64ad6 19764c72aca8152a51003bc396bce21a56c29306bccc36a37d816de2568e7794 a8f80d857132a3dbaf3ef75409cace85e1b85a8233ef2757a507158bbba8f5f6 d9c08ab4ee39f33a1c1a41ff2b8e81320050b6c4c167596b93918824a65fe1b0 6e7dcf5d7e128a77e5509f3a3a2af6f71de243cccfa5788088d277175e59bd61 a3b5cdc10cfd232cc14682d3b37d3aca68df524af6f38511e5627286034a7c08 1c7596bfc3adecc7d1868cb9d645f8cb61041bccad92249142542934977000c1 bfbbaf570bf1a3fad2368d1e6480ed485642acd1d44258a885562397035dc979 b08877eadba7c99963e7d866be89270b10629964b5c88439c08a7721e2fd6a7d 2a03d9cf2d7fa713411824db51a9141a7c3ca37feeb758ab37376d75b9ad8534 6fe75968a1e7f6b9b549b7bbef5da78116a7d029e6e95db78593abf789ab985e 301f5945c1628f591b4867079526cd9c779b05cd30de4d9dbb820ce964cb3acf cf693b39b15335996a208373a9a1cd362e93d740535d4fe06aaef81a77034010 883c098f8fe19617d6a3cc3ccaa63742d3b251b8baa94aba7ebb7f3cad53c3c5 2c626e3918191af1e343757e0fc5f9fa620ce7e655482aaab7303710167e3041 148e3e6aae44ebe904d96a7dea9281fdbdf93b03ccd08a315df83531214f1726 dd6eff4d5f494262983201ee2afd6cc77eac76155fc59d9e42a5c45735a37af6 c3c9c4aa7c535717838a1ce8ca3424188a3ddc2bb7f58483ef08fe9b6ea090c8 bd5a639bb56a54bed34a443f0f1624f13fffd6b4f24024573a6cdc110f2a85b3 f669c706b0bcf205ee9ff8342b38525e9cbf1f1b772a71c7718b30bd59f46490 dd92c121c9fd0b99edf33c57611ce19ca35fe17079c90ce99afdfd9227184562 f4e2dc14aea6d0989a8f01f17c0983ae121614d8f578d34ead9c05a88007bc4f 90e135d97abf57c1e32fd94ed63ceeddab6c1d42f3238e6cf85a063d1239a1d9 c5de6f9e7954b873ccef98d8b44f1761e66da28b9759a24f181c9f7ca8ca190d 4d5a52b1286ad43b5f8caebe64283da13e6a9e66536a821ebc1b50e98073d79e 335c244c2b8efcd7c0b223983db1608d6f1837ae6f73d970e567960e7e03d1a6 5f7dcaeecd4f9afd0266592ed051f800ee3d5a8954460f3f57a9fb9716f4dca2 616d5e77e19918f2b1c290fa9d049267df21a4b33a06e29c758e0ecac9746be0 b7988da1634f2e7162c2c483f1c8d78eaefc94e907d3bcc79cc389979ab0dc15 bf1b2371fe907e9c6f46e26260566e38ed55175ce116e4da7024db03b6657b44 1543e575e850440d7613b96baa8320e2cde5ab834d1f333a9a749c3ab079546c 3429c90c61bfda4b758861c2e351866a18c4d14d3bee4d863edce298231a71dc 9e4c701ac2d01fead65c2244a81865874b1c22125e9f11720450c10366eb6eec 694ba3acd45ea6f9cda354cc938222a523494fa570ba76d359afd28f6ca82c35 dc9890f6e09346d93734ef8c82d7b62344fb8103a1fe6f84fb56f55e1969c4bb f61c916f5fc7aaf919d620ed2c25d2bac648ede51f981afe1d4ecf8263830258 75069803a3032b0fba0522 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR17 %!PS-AdobeFont-1.1: CMR17 1.0 %%CreationDate: 1991 Aug 20 16:38:24 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR17) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR17 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 44 /comma put dup 45 /hyphen put dup 75 /K put dup 76 /L put dup 77 /M put dup 80 /P put dup 83 /S put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 104 /h put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 121 /y put readonly def /FontBBox{-33 -250 945 749}readonly def /UniqueXX 5000795 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486be356c29c3aec91ef17f5d31183b1e489f1fa 559a6693fbdd04d4f7e99886d8cb7cca13d4aa9ceb34a708af22e69ccef0e504 c3997075243585edf60b6447c8d01cffb04b6e3ddbaa7da5891286861322a795 e9d990fd56af2f5c2b35801a0b07874e4ac3170481d232e150443253c92a8b05 513d4a4154a99d91f629ccd30376500f5d16aaa9b203bad6350e4b19424c0e33 75f2949bc1927a6f6221252f8a96618e646aa1b0810753c98becc26b37837775 24bca72b818d314074c1c91f4e1f4148d3d703d12d179ee6970bd55d04aa9a9d 97f63f2c7dbd129ec2ed3ba5f8080270ea5854ea165dbcc061e2f47f1aaf0997 2ac9a689a2a9620b91b53f6b5092884372f5f7afca495f331138d85a97c5bb83 d8072dc88010fc89108b60bc06d4f93c722e8908d7ce3c1ea3c70e9e65edd769 7db17ce0e012ad37d4030f33d30a73fd655b7a2e572f3378ac933dd75082f2f1 68702be19bc60cef6bd5f0ada180d21408c04b7f7db9b6dbaefe09e809a80ddf 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont TeXDict begin 39158280 55380996 1000 600 600 (golden.dvi) @start /Fa 173[48 82[{}1 66.4176 /MSBM7 rf /Fb 145[38 54 1[33 25[48 21[19 25[39 6[32 13[35 12[{}8 49.8132 /CMMI6 rf /Fc 143[100 21[55 1[144 4[83 47 21[89 1[89 1[89 15[80 22[59 59 2[55 33 10[46 46{}15 99.6264 /CMEX10 rf /Fd 252[32 2[48{}2 49.8132 /CMSY6 rf /Fe 205[74 50[{}1 99.6264 /LASY10 rf /Ff 129[30 10[24 24 4[50 3[18 2[19 27 50[30 30 38[43 10[{}10 49.8132 /CMR6 rf /Fg 133[40 47 45 65 45 52 32 40 41 45 50 50 55 80 25 45 30 30 50 45 30 45 50 45 45 50 9[97 72 72 70 55 1[75 66 75 72 87 61 75 1[38 72 75 64 66 74 70 69 72 5[30 30 2[50 50 50 50 50 50 50 50 1[30 35 30 2[40 40 25[86 1[55 60 11[{}65 99.6264 /CMTI12 rf /Fh 177[57 72 77[{}2 66.4176 /EUFM7 rf /Fi 168[133 87[{}1 143.462 /CMMI12 rf /Fj 134[59 1[81 1[62 44 44 46 1[62 56 62 93 31 59 1[31 62 56 34 51 62 50 1[54 11[86 78 62 84 1[77 1[88 1[67 4[88 2[86 81 80 85 7[56 56 56 56 56 56 56 56 56 56 1[31 4[44 44 31 26[62 12[{}47 99.6264 /CMBX12 rf /Fk 129[35 3[31 6[28 28 2[35 1[59 20 2[20 1[35 22 31 1[31 5[20 1[20 3[72 4[52 16[53 3[55 8[35 35 35 35 35 4[55 1[27 27 29[51 55 51 8[{}29 66.4176 /CMR8 rf /Fl 133[33 35 40 3[25 33 32 3[43 62 1[37 1[24 4[36 9[48 5[41 43 1[56 4[48 60 39 1[58 5[53 53 2[55 35 1[20 20 24[44 46 1[42 1[31 1[36 1[31 1[42 41 3[35 1[29 31 1[40 45 11[{}38 66.4176 /CMMI8 rf /Fm 173[72 3[72 10[72 67[{}3 99.6264 /MSBM10 rf /Fn 148[50 28 2[50 50 9[66 66 72 67 71 2[62 9[76 2[84 3[77 52 65 2[77 7[0 0 2[66 66 100 4[100 2[100 100 2[100 1[50 1[100 5[77 77 4[77 77 5[50 3[77 1[77 1[77 2[50 1[28 77{}38 99.6264 /CMSY10 rf /Fo 133[45 48 55 1[47 1[35 46 44 2[47 58 85 29 2[33 2[48 45 1[42 1[51 6[67 2[92 1[66 57 60 1[77 63 1[78 94 66 83 54 43 81 77 63 72 81 70 74 73 2[76 49 76 27 27 24[61 63 1[58 1[42 55 50 55 43 1[59 57 5[39 43 51 55 12[{}56 99.6264 /CMMI12 rf /Fp 165[60 11[83 105 77[{}3 99.6264 /EUFM10 rf /Fq 252[42 3[{}1 83.022 /CMSY10 rf /Fr 168[111 60[60 26[{}2 119.552 /CMMI12 rf /Fs 131[134 1[60 71 71 97 71 75 52 53 55 1[75 67 75 112 37 71 1[37 75 67 41 61 75 60 75 65 9[139 102 103 94 75 100 1[92 101 1[128 81 105 1[50 2[85 2[97 96 102 7[67 67 67 67 67 67 67 67 67 67 1[37 45 5[37 24[112 1[75 12[{}56 119.552 /CMBX12 rf /Ft 133[72 85 2[85 90 63 64 66 1[90 81 90 134 45 2[45 90 81 49 74 90 72 90 78 12[112 90 120 1[110 2[153 1[126 1[60 1[127 101 3[115 122 11[81 81 81 81 81 3[54 45[{}38 143.462 /CMBX12 rf /Fu 128[49 49 1[98 49 43 51 51 70 51 54 38 38 38 51 54 49 54 81 27 51 30 27 54 49 30 43 54 43 54 49 3[27 1[27 60 73 1[100 73 73 70 54 72 76 66 76 73 89 61 76 50 35 73 77 64 66 75 70 69 73 3[76 1[27 27 49 49 49 49 49 49 49 49 49 49 49 27 33 27 76 1[38 38 27 4[49 13[49 49 3[81 81 54 54 57 70 76 70 5[76 81 61{}92 99.6264 /CMR12 rf /Fv 178[95 77[{}1 90.9091 /EUFM10 rf /Fw 148[35 20 30[54 11[55 12[47 71 19 12[35 1[71 11[55 55 13[55 2[35 1[20 55{}15 66.4176 /CMSY8 rf /Fx 168[86 58[40 15[51 12[{}3 90.9091 /CMMI10 rf /Fy 134[48 48 66 48 51 35 36 36 48 51 45 51 76 25 48 1[25 51 45 28 40 51 40 51 45 9[93 2[66 51 5[83 57 71 1[33 1[71 5[68 18[25 30 25 31[51 12[{}37 90.9091 /CMR10 rf /Fz 139[41 41 43 14[46 58 51 31[79 65[{}7 90.9091 /CMBX10 rf /FA 174[66 21[23 59[{}2 83.022 /CMMI10 rf /FB 203[42 42 42 42 49[{}4 83.022 /CMR10 rf /FC 133[52 62 62 85 62 65 46 46 46 1[65 59 65 98 33 62 1[33 65 1[36 52 65 52 65 59 1[33 5[88 1[120 88 88 85 65 86 91 80 1[88 107 73 91 60 1[88 92 76 1[89 85 83 88 7[59 59 59 59 59 59 59 59 59 59 1[33 39 33 23[59 59 8[85 10[{}61 119.552 /CMR12 rf /FD 252[60 3[{}1 119.552 /CMSY10 rf /FE 168[159 87[{}1 172.188 /CMMI12 rf /FF 134[83 2[83 88 61 62 61 2[79 88 133 43 2[43 88 1[47 70 88 70 88 79 13[88 2[108 2[146 99 124 29[52 43 44[{}25 172.188 /CMR17 rf end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin %%BeginPaperSize: a4 a4 %%EndPaperSize %%EndSetup %%Page: 1 1 1 0 bop 1201 809 a FF(P)l(erturbation)52 b(theory)289 1017 y(of)g FE(W)649 955 y 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y(represen)m(tativ)m(e)35 b(of)e Fo(!)t Fu(.)0 1542 y Fj(Prop)s(osition)38 b(2.4)49 b Fg(Ther)-5 b(e)34 b(exists)g(a)h(unique)g(r)-5 b(epr)g(esentation)1222 1762 y Fu(Aut)1387 1777 y Fl(!)1437 1762 y Fu(\()p Fp(M)p Fu(\))27 b Fn(3)i Fo(\032)f Fn(7!)f Fo(U)2021 1721 y Fk(\012)2077 1762 y Fu(\()p Fo(\032)p Fu(\))h Fn(2)g(U)10 b Fu(\()p Fn(H)q Fu(\))0 1982 y Fg(such)35 b(that)1040 2103 y Fo(U)1116 2061 y Fk(\012)1172 2103 y Fu(\()p Fo(\032)p Fu(\)\012)28 b(=)g(\012)p Fo(;)156 b(U)1829 2061 y Fk(\012)1885 2103 y Fu(\()p Fo(\032)p Fu(\))p Fo(AU)2160 2061 y Fk(\012)2216 2103 y Fu(\()p Fo(\032)p Fu(\))2342 2061 y Fw(\003)2410 2103 y Fu(=)27 b Fo(\032)p Fu(\()p Fo(A)p Fu(\))p Fo(:)0 2277 y Fg(It)36 b(is)h(c)-5 b(ontinuous)36 b(if)g(we)g(e)-5 b(quip)36 b Fu(Aut)1366 2292 y Fl(!)1416 2277 y Fu(\()p Fp(M)p Fu(\))g Fg(with)g(the)h(p)-5 b(ointwise)35 b Fo(\033)t Fg(-we)-5 b(ak)36 b(top)-5 b(olo)g(gy)36 b(and)g Fn(U)10 b Fu(\()p Fn(H)q Fu(\))37 b Fg(with)0 2397 y(the)e(str)-5 b(ong)35 b(op)-5 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b(A)27 b Fn(2)h Fp(M)p Fo(:)0 4405 y Fs(2.8)135 b(The)45 b(T)-11 b(omita-T)g(ak)l(esaki)46 b(theory)0 4589 y Fu(Let)38 b Fo(!)i Fu(b)s(e)e(a)f(faithful)g(state)h(on)f Fp(M)p Fu(.)58 b(By)38 b(passing)g(to)f(the)h(GNS)f(represen)m(tation)i(w)m(e) g(ma)m(y)f(assume)0 4710 y(that)32 b Fp(M)27 b Fn(\032)h(B)s Fu(\()p Fn(H)q Fu(\))33 b(and)f(that)f Fo(!)36 b Fu(has)c(a)g(v)m (ector)h(represen)m(tativ)m(e)i(\012)d(whic)m(h)i(is)e(cyclic)i(and)e (separating.)146 4830 y(The)48 b(follo)m(wing)f(theorem)h(summarizes)h (the)e(results)i(of)d(the)h(w)m(ell)i(kno)m(wn)f(T)-8 b(omita-T)g(ak)m(esaki)0 4951 y(theory)g(.)1865 5753 y(9)p eop %%Page: 10 10 10 9 bop 0 407 a Fj(Theorem)75 b(2.6)90 b Fu(\(1\))49 b Fg(De\014ne)34 b(the)g(op)-5 b(er)g(ator)35 b Fo(S)1832 422 y Fk(\012)1922 407 y Fg(with)g(the)g(domain)e Fp(M)p Fu(\012)i Fg(by)1697 627 y Fo(S)1757 642 y Fk(\012)1812 627 y Fo(A)p Fu(\012)28 b(=)g Fo(A)2160 586 y Fw(\003)2199 627 y Fu(\012)p Fo(:)214 847 y Fg(Then)40 b Fo(S)534 862 y Fk(\012)630 847 y Fg(is)g(antiline)-5 b(ar,)41 b(closable,)f(has)g(a)g(zer)-5 b(o)40 b(kernel)g(and)g(c)-5 b(okernel.)60 b(Its)40 b(closur)-5 b(e)40 b(wil)5 b(l)40 b(b)-5 b(e)214 967 y(denote)g(d)35 b(also)f Fo(S)830 982 y Fk(\012)885 967 y Fg(.)45 b(L)-5 b(et)35 b Fo(S)1188 982 y Fk(\012)1271 967 y Fu(=)27 b Fo(J)9 b Fu(\001)1518 916 y Fk(1)p Fl(=)p Fk(2)1518 994 y(\012)1664 967 y Fg(b)-5 b(e)34 b(its)h(p)-5 b(olar)34 b(de)-5 b(c)g(omp)g(osition;)41 1129 y Fu(\(2\))48 b Fo(J)c Fg(is)35 b(an)g(antiunitary)g(involution;) 41 1291 y Fu(\(3\))48 b(\001)295 1306 y Fk(\012)386 1291 y Fg(is)34 b(a)h(p)-5 b(ositive)34 b(op)-5 b(er)g(ator)35 b(satisfying)f Fo(J)9 b Fu(\001)1887 1306 y Fk(\012)1943 1291 y Fo(J)37 b Fu(=)27 b(\001)2218 1250 y Fw(\000)p Fk(1)2218 1318 y(\012)2348 1291 y Fg(and)34 b Fu(\001)2618 1306 y Fk(\012)2673 1291 y Fu(\012)28 b(=)g(\012)p Fg(;)41 1453 y Fu(\(4\))48 b Fg(The)35 b(map)1252 1573 y Fo(\034)1305 1532 y Fl(t)1294 1598 y(!)1345 1573 y Fu(\()p Fo(A)p Fu(\))27 b(:=)h(\001)1733 1532 y Fw(\000)p Fk(i)p Fl(t)1837 1573 y Fo(A)p Fu(\001)1991 1532 y Fk(i)p Fl(t)2069 1573 y Fn(2)g Fp(M)p Fo(;)121 b(A)27 b Fn(2)i Fp(M)p Fo(;)214 1748 y Fg(is)35 b(a)g Fo(W)510 1711 y Fw(\003)549 1748 y Fg(-dynamics)f(on)g Fp(M)h Fg(and)f Fn(\000)17 b Fu(log)h(\001)1801 1763 y Fk(\012)1891 1748 y Fg(is)34 b(its)h Fu(\012)p Fg(-Liouvil)5 b(le)-5 b(an.)146 1951 y Fu(The)36 b Fo(W)455 1915 y Fw(\003)494 1951 y Fu(-dynamics)g Fm(R)31 b Fn(3)h Fo(t)f Fn(7!)g Fo(\034)1411 1915 y Fw(\000)p Fl(t)1400 1976 y(!)1531 1951 y Fu(is)k(called)g(the)h(mo)s(dular)e(dynamics)j (and)d(\001)3175 1966 y Fk(\012)3265 1951 y Fu(is)i(called)f(the)0 2071 y(mo)s(dular)d(op)s(erator.)0 2360 y Fs(2.9)135 b(Standard)45 b(form)0 2545 y Fu(One)31 b(of)e(the)i(cen)m(tral)g (notions)f(of)g(the)h(theory)g(of)e Fo(W)1930 2509 y Fw(\003)1969 2545 y Fu(-algebras)h(is)h(the)g(so-called)f(standard)h (form.)42 b(It)0 2665 y(has)d(b)s(een)g(in)m(tro)s(duced)h(b)m(y)g (Haagerup)e([Haa],)i(follo)m(wing)f(the)g(w)m(ork)h(of)e(Araki)h([Ar3]) f(and)h(Connes)0 2786 y([Co].)146 2906 y(A)33 b Fo(W)358 2870 y Fw(\003)397 2906 y Fu(-algebra)g(in)g(a)f(standard)h(form)g(is)g (a)g(quadruple)h(\()p Fp(M)p Fo(;)17 b Fn(H)q Fo(;)g(J)n(;)g Fn(H)2740 2870 y Fk(+)2798 2906 y Fu(\),)33 b(where)h Fn(H)g Fu(is)g(a)e(Hilb)s(ert)0 3026 y(space,)h Fp(M)27 b Fn(\032)i(B)s Fu(\()p Fn(H)q Fu(\))j(is)g(a)g(concrete)h Fo(W)1454 2990 y Fw(\003)1493 3026 y Fu(-algebra,)e Fo(J)41 b Fu(is)33 b(an)f(an)m(tiunitary)g(in)m(v)m(olution)i(on)d Fn(H)i Fu(\(that)f(is,)0 3147 y Fo(J)42 b Fu(is)33 b(an)m(tilinear,)g Fo(J)720 3111 y Fk(2)787 3147 y Fu(=)28 b(1,)k Fo(J)1062 3111 y Fw(\003)1130 3147 y Fu(=)27 b Fo(J)9 b Fu(\))33 b(and)f Fn(H)1641 3111 y Fk(+)1733 3147 y Fu(is)h(a)f(self-dual)h(cone) h(in)e Fn(H)i Fu(suc)m(h)g(that:)0 3267 y(\(1\))e Fo(J)9 b Fp(M)p Fo(J)37 b Fu(=)27 b Fp(M)624 3231 y Fw(0)647 3267 y Fu(;)0 3388 y(\(2\))32 b Fo(J)9 b(AJ)37 b Fu(=)28 b Fo(A)561 3351 y Fw(\003)633 3388 y Fu(for)k Fo(A)h Fu(in)f(the)h(cen)m(ter)h(of)e Fp(M)p Fu(;)0 3508 y(\(3\))g Fo(J)9 b Fu(\011)28 b(=)f(\011)33 b(for)f(\011)27 b Fn(2)h(H)967 3472 y Fk(+)1027 3508 y Fu(;)0 3628 y(\(4\))k Fo(AJ)9 b(A)p Fn(H)451 3592 y Fk(+)538 3628 y Fn(\032)29 b(H)729 3592 y Fk(+)820 3628 y Fu(for)j Fo(A)c Fn(2)g Fp(M)p Fu(.)146 3749 y(If)41 b Fp(M)g Fu(is)g(an)g(abstract)h Fo(W)1143 3713 y Fw(\003)1182 3749 y Fu(-algebra,)g(then)g(w)m(e)g (will)g(sa)m(y)g(that)f(\()p Fo(\031)t(;)17 b Fn(H)q Fo(;)g(J)n(;)g Fn(H)3017 3713 y Fk(+)3075 3749 y Fu(\))41 b(is)h(its)f(standard)0 3869 y(represen)m(tation)27 b(if)e Fo(\031)31 b Fu(:)d Fp(M)f Fn(!)h(B)s Fu(\()p Fn(H)q Fu(\))d(is)h(an)e(injectiv)m(e)j(unital)f(represen)m(tation)h(and)e(\() p Fo(\031)t Fu(\()p Fp(M)p Fu(\))p Fo(;)17 b Fn(H)q Fo(;)g(J)n(;)g Fn(H)3684 3833 y Fk(+)3742 3869 y Fu(\))0 3990 y(is)33 b(a)f(standard)h(form.)0 4218 y Fj(Theorem)75 b(2.7)49 b Fg(L)-5 b(et)48 b Fp(M)f Fg(b)-5 b(e)47 b(a)g Fo(W)1372 4182 y Fw(\003)1411 4218 y Fg(-algebr)-5 b(a)46 b(with)h(a)g(faithful)g (state)g Fo(!)t Fg(.)81 b(L)-5 b(et)48 b Fo(\031)54 b Fu(:)d Fp(M)f Fn(!)g(B)s Fu(\()p Fn(H)q Fu(\))0 4338 y Fg(b)-5 b(e)37 b(the)h(c)-5 b(orr)g(esp)g(onding)36 b(GNS)i(r)-5 b(epr)g(esentation)36 b(with)i(the)f(cyclic)g(ve)-5 b(ctor)38 b Fu(\012)p Fg(.)53 b(L)-5 b(et)38 b Fo(J)46 b Fg(b)-5 b(e)38 b(the)f(mo)-5 b(dular)0 4459 y(c)g(onjugation)25 b(obtaine)-5 b(d)26 b(by)g(the)g(T)-7 b(omita-T)g(akesaki)23 b(the)-5 b(ory)27 b(and)e Fn(H)2446 4422 y Fk(+)2533 4459 y Fu(:=)j Fn(f)p Fo(\031)t Fu(\()p Fo(A)p Fu(\))p Fo(J)9 b(\031)t Fu(\()p Fo(A)p Fu(\)\012)27 b(:)h Fo(A)g Fn(2)g Fp(M)p Fn(g)3695 4422 y Fk(cl)3750 4459 y Fg(.)0 4579 y(Then)46 b Fn(H)351 4543 y Fk(+)457 4579 y Fg(is)h(a)f(self-dual) h(c)-5 b(one)46 b(and)g Fu(\()p Fo(\031)t(;)17 b Fn(H)q Fo(;)g(J)n(;)g Fn(H)1957 4543 y Fk(+)2016 4579 y Fu(\))46 b Fg(is)h(a)g(standar)-5 b(d)46 b(r)-5 b(epr)g(esentation)46 b(of)h Fp(M)p Fg(.)80 b(If)0 4699 y Fu(\()p Fo(\031)t(;)17 b Fn(H)q Fo(;)g(J)324 4714 y Fk(1)363 4699 y Fo(;)g Fn(H)492 4658 y Fk(+)491 4724 y(1)551 4699 y Fu(\))34 b Fg(is)f(another)h (standar)-5 b(d)34 b(r)-5 b(epr)g(esentation)33 b(of)h Fp(M)g Fg(and)f Fu(\012)28 b Fn(2)g(H)2832 4658 y Fk(+)2831 4724 y(1)2891 4699 y Fg(,)35 b(then)e Fn(H)3256 4658 y Fk(+)3255 4724 y(1)3343 4699 y Fu(=)28 b Fn(H)3532 4658 y Fk(+)3531 4724 y(2)3625 4699 y Fg(and)0 4820 y Fo(J)54 4835 y Fk(1)121 4820 y Fu(=)g Fo(J)279 4835 y Fk(2)318 4820 y Fg(.)0 5048 y Fj(Theorem)75 b(2.8)49 b Fg(Every)41 b Fo(W)1095 5012 y Fw(\003)1134 5048 y Fg(-algebr)-5 b(a)39 b Fp(M)g Fg(p)-5 b(ossesses)39 b(a)h(standar)-5 b(d)39 b(r)-5 b(epr)g(esentation.)60 b(Mor)-5 b(e)g(over,)41 b(if)0 5168 y Fu(\()p Fo(\031)93 5183 y Fk(1)133 5168 y Fo(;)17 b Fn(H)261 5183 y Fk(1)300 5168 y Fo(;)g(J)398 5183 y Fk(1)437 5168 y Fo(;)g Fn(H)566 5127 y Fk(+)565 5193 y(1)625 5168 y Fu(\))41 b Fg(and)g Fu(\()p Fo(\031)993 5183 y Fk(2)1033 5168 y Fo(;)17 b Fn(H)1161 5183 y Fk(2)1200 5168 y Fo(;)g(J)1298 5183 y Fk(2)1337 5168 y Fo(;)g Fn(H)1466 5127 y Fk(+)1465 5193 y(2)1525 5168 y Fu(\))41 b Fg(ar)-5 b(e)41 b(two)h(standar)-5 b(d)40 b(r)-5 b(epr)g(esentations)41 b(of)g Fp(M)p Fg(,)h(then)g(ther)-5 b(e)1841 5753 y Fu(10)p eop %%Page: 11 11 11 10 bop 0 407 a Fg(exists)35 b(a)f(unique)h(unitary)g(op)-5 b(er)g(ator)35 b Fo(W)1497 371 y Fw(0)1548 407 y Fu(:)27 b Fn(H)1686 422 y Fk(1)1754 407 y Fn(!)g(H)1965 422 y Fk(2)2039 407 y Fg(such)35 b(that)1438 598 y Fo(W)1544 557 y Fw(0)1567 598 y Fo(\031)1622 613 y Fk(1)1662 598 y Fu(\()p Fo(A)p Fu(\))27 b(=)h Fo(\031)1997 613 y Fk(2)2037 598 y Fu(\()p Fo(A)p Fu(\))p Fo(W)2292 557 y Fw(0)2315 598 y Fo(;)1538 814 y(W)1644 773 y Fw(0)1666 814 y Fn(H)1751 773 y Fk(+)1750 839 y(1)1838 814 y Fu(=)g Fn(H)2027 773 y Fk(+)2026 839 y(2)2086 814 y Fo(:)0 1011 y Fg(We)35 b(then)g(automatic)-5 b(al)5 b(ly)34 b(have)h Fo(W)1327 975 y Fw(0)1350 1011 y Fo(J)1404 1026 y Fk(1)1471 1011 y Fu(=)27 b Fo(J)1628 1026 y Fk(2)1667 1011 y Fo(W)1773 975 y Fw(0)1796 1011 y Fg(.)146 1217 y Fu(If)35 b Fp(M)f Fu(is)h Fo(\033)t Fu(-\014nite,)g(then)g(Theorem)h(2.8)e(is)h(pro)m(v)m (en)h(e.g.)50 b(in)34 b([BR1].)50 b(In)34 b(this)i(case)f(the)g (existence)0 1337 y(part)d(follo)m(ws)i(from)e(Theorem)i(2.7.)146 1458 y(If)c Fp(M)e Fu(is)i(not)f Fo(\033)t Fu(-\014nite,)i(the)e (theorem)h(is)g(pro)m(v)m(en)h(using)f(w)m(eigh)m(ts)h(instead)f(of)f (states.)43 b(The)31 b(details)0 1578 y(can)i(b)s(e)g(found)f(in)h ([Haa,)f(St)q(].)0 1864 y Fs(2.10)136 b(States)46 b(and)e (automorphisms)i(in)f(the)g(standard)g(represen)l(tation)0 2049 y Fu(In)33 b(this)g(subsection)i(w)m(e)e(\014x)h(a)e Fo(W)1254 2012 y Fw(\003)1293 2049 y Fu(-algebra)g(in)h(the)g(standard) g(form)f(\()p Fp(M)p Fo(;)17 b Fn(H)q Fo(;)g(J)n(;)g Fn(H)3083 2012 y Fk(+)3141 2049 y Fu(\).)0 2254 y Fj(Theorem)75 b(2.9)90 b Fu(\(1\))1551 2374 y Fn(H)1636 2333 y Fk(+)1723 2374 y Fn(3)28 b Fu(\012)g Fn(7!)f Fo(!)2103 2389 y Fk(\012)2186 2374 y Fn(2)h Fp(M)2385 2333 y Fk(+)2385 2399 y Fw(\003)214 2540 y Fg(is)35 b(a)g(bije)-5 b(ction.)44 b(Its)34 b(inverse)g(wil)5 b(l)35 b(b)-5 b(e)34 b(denote)-5 b(d)1538 2740 y Fp(M)1643 2699 y Fk(+)1643 2765 y Fw(\003)1729 2740 y Fn(3)28 b Fo(!)j Fn(7!)c Fu(\012)2112 2755 y Fl(!)2191 2740 y Fn(2)h(H)2370 2699 y Fk(+)2429 2740 y Fo(:)41 2978 y Fu(\(2\))48 b Fg(If)35 b Fu(\011)p Fo(;)17 b Fu(\010)27 b Fn(2)i(H)714 2942 y Fk(+)773 2978 y Fg(,)34 b(then)1052 3178 y Fn(k)p Fu(\011)21 b Fn(\000)i Fu(\010)p Fn(k)1419 3137 y Fk(2)1486 3178 y Fn(\024)29 b(k)p Fo(!)1703 3193 y Fk(\011)1783 3178 y Fn(\000)23 b Fo(!)1944 3193 y Fk(\010)1999 3178 y Fn(k)k(\024)h(k)p Fu(\011)22 b Fn(\000)h Fu(\010)p Fn(kk)p Fu(\011)f(+)g(\010)p Fn(k)p Fo(:)41 3417 y Fu(\(3\))48 b Fg(If)35 b Fu(\012)28 b Fn(2)g(H)594 3381 y Fk(+)653 3417 y Fg(,)35 b(then)g Fu(\012)g Fg(is)g(cyclic)f Fn(,)g Fu(\012)i Fg(is)e(sep)-5 b(ar)g(ating)34 b Fn(,)h Fo(!)2418 3432 y Fk(\012)2508 3417 y Fg(is)f(faithful.)41 3576 y Fu(\(4\))48 b Fg(F)-7 b(or)34 b Fu(\012)29 b Fn(2)f(H)674 3540 y Fk(+)733 3576 y Fg(,)34 b Fu(s)835 3540 y Fw(0)835 3601 y Fk(\012)919 3576 y Fu(=)27 b Fo(J)9 b Fu(s)1123 3591 y Fk(\012)1179 3576 y Fo(J)g Fg(.)146 3781 y Fu(The)34 b(v)m(ector)f(\012)709 3796 y Fl(!)788 3781 y Fn(2)28 b(H)967 3745 y Fk(+)1059 3781 y Fu(will)33 b(b)s(e)g(called)g(the)g Fg(standar)-5 b(d)34 b(ve)-5 b(ctor)35 b(r)-5 b(epr)g(esentative)34 b(of)h Fo(!)t Fg(.)146 3902 y Fu(A)e(unitary)g(op)s(erator)f Fo(U)43 b Fu(on)33 b Fn(H)g Fu(is)g(called)h(a)e(standard)h(unitary)g (op)s(erator)f(i\013)0 4022 y(\(1\))g Fo(U)10 b Fn(H)318 3986 y Fk(+)406 4022 y Fu(=)27 b Fn(H)594 3986 y Fk(+)653 4022 y Fu(,)0 4142 y(\(2\))32 b Fo(U)10 b Fp(M)p Fo(U)414 4106 y Fw(\003)482 4142 y Fu(=)28 b Fp(M)p Fu(.)0 4348 y Fj(Theorem)75 b(2.10)91 b Fu(\(1\))48 b Fg(If)31 b Fo(U)43 b Fg(is)32 b(a)g(standar)-5 b(d)31 b(unitary)i(op)-5 b(er)g(ator,)32 b(then)g Fo(J)9 b(U)38 b Fu(=)28 b Fo(U)10 b(J)42 b Fg(and)31 b Fo(U)10 b Fp(M)3536 4312 y Fw(0)3560 4348 y Fo(U)3636 4312 y Fw(\003)3704 4348 y Fu(=)214 4468 y Fp(M)319 4432 y Fw(0)342 4468 y Fg(.)41 4627 y Fu(\(2\))48 b Fg(Ther)-5 b(e)35 b(exists)f(a)h(unique)g(unitary)g(r)-5 b(epr)g(esentation)1382 4827 y Fu(Aut\()p Fp(M)p Fu(\))27 b Fn(3)h Fo(\032)g Fn(7!)f Fo(U)10 b Fu(\()p Fo(\032)p Fu(\))29 b Fn(2)f(U)10 b Fu(\()p Fn(H)q Fu(\))967 b(\(2.4\))214 5027 y Fg(satisfying)35 b(the)g(fol)5 b(lowing)33 b(c)-5 b(onditions:)214 5147 y Fu(\(a\))35 b Fo(U)10 b Fu(\()p Fo(\032)p Fu(\))p Fo(AU)g Fu(\()p Fo(\032)p Fu(\))851 5111 y Fw(\003)920 5147 y Fu(=)27 b Fo(\032)p Fu(\()p Fo(A)p Fu(\))p Fg(,)36 b Fo(A)27 b Fn(2)h Fp(M)p Fg(;)214 5268 y Fu(\(b\))35 b Fo(U)10 b Fu(\()p Fo(\032)p Fu(\))p Fn(H)666 5232 y Fk(+)754 5268 y Fn(\032)28 b(H)944 5232 y Fk(+)1003 5268 y Fg(.)1841 5753 y Fu(11)p eop %%Page: 12 12 12 11 bop 41 407 a Fu(\(3\))48 b Fg(The)35 b(image)f(of)g(\(2.4\))g(is) h(the)g(gr)-5 b(oup)35 b(of)f(al)5 b(l)35 b(standar)-5 b(d)34 b(unitary)h(op)-5 b(er)g(ators.)41 569 y Fu(\(4\))48 b Fg(\(2.4\))43 b(is)h(c)-5 b(ontinuous)43 b(if)g Fu(Aut\()p Fp(M)p Fu(\))g Fg(is)h(e)-5 b(quipp)g(e)g(d)43 b(with)g(the)h(p)-5 b(ointwise)42 b Fo(\033)t Fg(-we)-5 b(ak)43 b(top)-5 b(olo)g(gy)43 b(and)214 689 y Fn(U)10 b Fu(\()p Fn(H)q Fu(\))36 b Fg(with)e(the)h(str)-5 b(ong)35 b(op)-5 b(er)g(ator)34 b(top)-5 b(olo)g(gy.)41 851 y Fu(\(5\))48 b Fo(U)10 b Fu(\()p Fo(\032)p Fu(\)\012)486 866 y Fl(!)566 851 y Fu(=)27 b(\012)739 868 y Fl(\032)775 849 y Fd(\000)p Ff(1)p Fd(\003)890 868 y Fl(!)976 851 y Fg(for)34 b(al)5 b(l)35 b Fo(!)c Fn(2)d Fp(M)1562 815 y Fk(+)1562 876 y Fw(\003)1620 851 y Fg(.)146 1079 y Fo(U)10 b Fu(\()p Fo(\032)p Fu(\))34 b(will)f(b)s(e)g(called)g(the)g(standard)g(implemen) m(tation)i(of)d Fo(\032)p Fu(.)146 1200 y(Supp)s(ose)39 b(that)f Fo(t)e Fn(!)g Fo(\034)1011 1164 y Fl(t)1079 1200 y Fu(is)i(a)g Fo(W)1375 1164 y Fw(\003)1414 1200 y Fu(-dynamics)h(on)f Fp(M)f Fu(and)h(let)g Fo(U)10 b Fu(\()p Fo(\034)2674 1164 y Fl(t)2705 1200 y Fu(\))37 b(b)s(e)h(as)g(in)g(Theorem)h(2.10.)0 1320 y(Then)34 b(there)f(exists)h(a)f(unique)h(self-adjoin)m(t)f Fo(L)f Fu(suc)m(h)i(that)1622 1540 y Fo(U)10 b Fu(\()p Fo(\034)1789 1499 y Fl(t)1820 1540 y Fu(\))28 b(=)f(e)2032 1499 y Fk(i)p Fl(tL)2130 1540 y Fo(:)0 1760 y Fu(The)35 b(op)s(erator)e Fo(L)h Fu(will)g(b)s(e)g(called)h(the)f(standard)g(Liouvillean)h(of)e (the)h Fo(W)2761 1724 y Fw(\003)2800 1760 y Fu(-dynamics)h Fo(\034)11 b Fu(,)35 b(or)e(simply)0 1880 y(the)g(Liouvillean)h(of)e Fo(\034)11 b Fu(.)0 2109 y Fj(Theorem)75 b(2.11)49 b Fg(The)35 b(Liouvil)5 b(le)-5 b(an)34 b(of)g Fo(\034)47 b Fg(is)34 b(the)h(unique)g(self-adjoint)f(op)-5 b(er)g(ator)34 b Fo(L)h Fg(satisfying)894 2329 y Fu(e)937 2288 y Fk(i)p Fl(tL)1034 2329 y Fn(H)1119 2288 y Fk(+)1206 2329 y Fn(\032)28 b(H)1396 2288 y Fk(+)1455 2329 y Fo(;)157 b Fu(e)1682 2288 y Fk(i)p Fl(tL)1779 2329 y Fo(A)p Fu(e)1895 2288 y Fw(\000)p Fk(i)p Fl(tL)2075 2329 y Fu(=)28 b Fo(\034)2232 2288 y Fl(t)2262 2329 y Fu(\()p Fo(A)p Fu(\))p Fo(;)121 b(A)28 b Fn(2)g Fp(M)p Fo(;)0 2549 y Fg(for)35 b(al)5 b(l)34 b Fo(t)28 b Fn(2)g Fm(R)p Fg(.)146 2777 y Fu(The)43 b(\014nal)f(result)h(w)m(e)g(wish)g(to)e(men)m(tion)i(follo)m(ws)g (easily)g(from)f(Theorems)h(2.9)f(and)g(2.10.)70 b(It)0 2897 y(has)38 b(b)s(een)f(a)g(k)m(ey)i(to)s(ol)d(in)i(recen)m(t)g(in)m (v)m(estigations)h(of)e(in)m(v)-5 b(arian)m(t)38 b(states)g(of)e(a)h (certain)h(class)g(of)f Fo(W)3708 2861 y Fw(\003)3747 2897 y Fu(-)0 3018 y(dynamical)d(systems)h(called)e(P)m(auli-Fierz)g (systems)i([BFS)q(,)d(DJ2,)g(JP1,)h(JP2,)g(M].)0 3246 y Fj(Theorem)75 b(2.12)49 b Fg(L)-5 b(et)36 b Fo(\034)46 b Fg(b)-5 b(e)35 b(a)f Fo(W)1327 3210 y Fw(\003)1366 3246 y Fg(-dynamics)g(and)g Fo(L)i Fg(the)f(c)-5 b(orr)g(esp)g(onding) 33 b(Liouvil)5 b(le)-5 b(an.)44 b(Then)689 3466 y Fn(f)p Fo(!)800 3481 y Fk(\010)882 3466 y Fu(:)28 b(\010)g Fn(2)g(H)1214 3425 y Fk(+)1295 3466 y Fn(\\)23 b Fu(Ker)p Fo(L)p Fn(g)28 b Fu(=)f Fn(f)p Fo(!)k Fn(2)d Fp(M)2129 3425 y Fk(+)2129 3491 y Fw(\003)2215 3466 y Fu(:)g Fo(!)73 b Fg(is)35 b Fo(\034)2562 3425 y Fl(t)2627 3466 y Fg(invariant)o Fn(g)p Fo(:)0 3686 y Fg(Conse)-5 b(quently,)41 3838 y Fu(\(1\))48 b(dim)18 b(Ker)p Fo(L)28 b Fu(=)f(0)35 b Fn(,)g Fg(ther)-5 b(e)34 b(ar)-5 b(e)35 b(no)f(normal)g Fo(\034)11 b Fg(-invariant)35 b(states.)41 4000 y Fu(\(2\))48 b(dim)18 b(Ker)p Fo(L)28 b Fu(=)f(1)35 b Fn(,)g Fg(ther)-5 b(e)34 b(exists)h(exactly)g(one)f(normal)g Fo(\034)11 b Fg(-invariant)34 b(state.)0 4228 y Fu(W)-8 b(e)33 b(will)g(not)g(mak) m(e)g(use)h(of)e(this)h(result)h(in)f(our)f(pap)s(er.)0 4517 y Fs(2.11)136 b(Comparison)0 4701 y Fu(In)44 b(some)h (circumstances)h(the)e(setups)i(of)d(Subsections)j(2.7)d(and)h(2.10)f (o)m(v)m(erlap.)79 b(Recall)44 b(that)f(in)0 4822 y(Subsection)g(2.7)e (w)m(e)h(ha)m(v)m(e)h(a)d Fo(W)1244 4786 y Fw(\003)1284 4822 y Fu(-algebra)g Fp(M)h Fu(with)h(a)f(faithful)g(state)g Fo(!)t Fu(.)69 b(W)-8 b(e)41 b(can)h(assume)h(that)0 4942 y Fp(M)27 b Fn(\032)h(B)s Fu(\()p Fn(H)q Fu(\))33 b(and)g(that)f Fo(!)k Fu(has)d(a)f(cyclic)j(v)m(ector)e(represen)m (tativ)m(e)j(\012.)146 5063 y(By)29 b(Theorem)g(2.7,)f(w)m(e)h(can)f (construct)h Fo(J)37 b Fu(and)28 b Fn(H)1980 5026 y Fk(+)2067 5063 y Fu(so)g(that)g(\()p Fp(M)p Fo(;)17 b Fn(H)q Fo(;)g(J)n(;)g Fn(H)2886 5026 y Fk(+)2944 5063 y Fu(\))28 b(is)g(a)f(standard)i(form)0 5183 y(and)k(\012)28 b Fn(2)g(H)467 5147 y Fk(+)526 5183 y Fu(.)1841 5753 y(12)p eop %%Page: 13 13 13 12 bop 0 407 a Fj(Prop)s(osition)38 b(2.13)49 b Fg(L)-5 b(et)45 b Fo(\032)i Fn(2)f Fu(Aut)1407 422 y Fl(!)1457 407 y Fu(\()p Fp(M)p Fu(\))p Fg(.)74 b(Supp)-5 b(ose)45 b(that)g Fo(U)57 b Fn(2)47 b(U)10 b Fu(\()p Fn(H)q Fu(\))45 b Fg(implements)f Fo(\032)p Fg(,)j(that)f(is)0 527 y Fo(\032)p Fu(\()p Fo(A)p Fu(\))28 b(=)g Fo(U)10 b(AU)556 491 y Fw(\003)596 527 y Fg(,)35 b Fo(A)28 b Fn(2)g Fp(M)p Fg(.)44 b(Then)34 b(the)h(fol)5 b(lowing)34 b(c)-5 b(onditions)34 b(ar)-5 b(e)34 b(e)-5 b(quivalent:)41 679 y Fu(\(1\))48 b Fo(U)10 b Fu(\012)29 b(=)f(\012)35 b(\()p Fo(U)j Fu(=)28 b Fo(U)920 643 y Fk(\012)976 679 y Fu(\()p Fo(\032)p Fu(\))35 b Fg(is)f(the)h Fu(\012)p Fg(-implementation)f(of)h Fo(\032)p Fu(\))p Fg(;)41 840 y Fu(\(2\))48 b Fo(U)10 b Fn(H)375 804 y Fk(+)463 840 y Fu(=)27 b Fn(H)651 804 y Fk(+)745 840 y Fu(\()p Fo(U)39 b Fu(=)27 b Fo(U)10 b Fu(\()p Fo(\032)p Fu(\))36 b Fg(is)f(the)g(standar)-5 b(d)34 b(implementation)f(of)i Fo(\032)p Fu(\))p Fg(.)0 1067 y Fj(Pro)s(of.)58 b Fu(W)-8 b(e)30 b(kno)m(w)g(from)f(Theorem)h (2.4)f(that)g(the)g(\012-implemen)m(tation)i(of)e Fo(\032)g Fu(exists)i(and)e(is)h(unique.)0 1188 y(W)-8 b(e)41 b(also)g(kno)m(w)g (from)g(Theorem)g(2.10)f(that)h(the)g(standard)g(implemen)m(tation)h (of)e Fo(\032)h Fu(exists)h(and)f(is)0 1308 y(unique.)k(Hence,)34 b(it)f(is)g(su\016cien)m(t)h(to)f(sho)m(w)g(the)g(implication)h(in)e (one)h(direction.)146 1429 y(\(2\))p Fn(\))p Fu(\(1\).)66 b(The)42 b(v)m(ector)f Fo(U)10 b Fu(\012)42 b(determines)g(the)f(state) g Fo(\032)2260 1392 y Fw(\003)2299 1429 y Fo(!)k Fu(=)c Fo(!)t Fu(.)66 b(Hence)42 b(the)f(v)m(ectors)h Fo(U)10 b Fu(\012,)43 b(\012)0 1549 y(b)s(elong)33 b(to)f(the)h(cone)g Fn(H)906 1513 y Fk(+)998 1549 y Fu(and)f(determine)j(the)e(same)g (state.)44 b(This)34 b(implies)g Fo(U)10 b Fu(\012)29 b(=)e(\012.)44 b Fe(2)146 1719 y Fu(As)25 b(a)f(corollary)-8 b(,)26 b(if)e(the)h(in)m(v)-5 b(arian)m(t)24 b(state)h Fo(!)i Fu(is)e(faithful,)h(then)f(the)f(concepts)i(of)e(the)g (\012-Liouvillean)0 1839 y(and)33 b(the)g(standard)g(Liouvillean)g (coincide.)0 2066 y Fj(Prop)s(osition)38 b(2.14)49 b Fg(L)-5 b(et)40 b Fo(t)d Fn(7!)f Fo(\034)1288 2030 y Fl(t)1358 2066 y Fg(b)-5 b(e)39 b(a)h Fo(W)1678 2030 y Fw(\003)1717 2066 y Fg(-dynamics)e(on)i Fp(M)f Fg(that)h(le)-5 b(aves)39 b(invariant)g(a)h(faithful)0 2186 y(state)35 b Fo(!)t Fg(.)46 b(Supp)-5 b(ose)34 b(that)i Fo(L)f Fg(is)g(a)g (self-adjoint)f(op)-5 b(er)g(ator)35 b(such)g(that)h Fo(\034)2602 2150 y Fl(t)2632 2186 y Fu(\()p Fo(A)p Fu(\))28 b(=)g(e)2956 2150 y Fk(i)p Fl(tL)3054 2186 y Fo(A)p Fu(e)3170 2150 y Fw(\000)p Fk(i)p Fl(tL)3322 2186 y Fg(.)46 b(Then)35 b(the)0 2307 y(fol)5 b(lowing)33 b(c)-5 b(onditions)34 b(ar)-5 b(e)35 b(e)-5 b(quivalent:)41 2458 y Fu(\(1\))48 b Fo(L)p Fu(\012)29 b(=)f(0)34 b(\()p Fo(L)28 b Fu(=)g Fo(L)868 2422 y Fk(\012)958 2458 y Fg(is)35 b(the)g Fu(\012)p Fg(-Liouvil)5 b(le)-5 b(an)34 b(of)g Fo(\034)11 b Fu(\))p Fg(;)41 2620 y Fu(\(2\))48 b Fg(F)-7 b(or)34 b Fo(t)28 b Fn(2)g Fm(R)p Fg(,)35 b Fu(e)733 2584 y Fk(i)p Fl(tL)831 2620 y Fn(H)916 2584 y Fk(+)1002 2620 y Fn(\032)29 b(H)1193 2584 y Fk(+)1287 2620 y Fu(\()p Fo(L)35 b Fg(is)f(the)h(standar)-5 b(d)35 b(Liouvil)5 b(le)-5 b(an)34 b(of)g Fo(\034)11 b Fu(\))p Fg(.)0 2940 y Fs(2.12)136 b(KMS)44 b(states)0 3125 y Fu(In)23 b(this)g(subsection)i(w)m(e)e(recall)h(basic)f(prop)s (erties)h(of)e(KMS)h(states.)41 b(Let)23 b(\()p Fp(M)p Fo(;)17 b(\034)2882 3088 y Fl(t)2911 3125 y Fu(\))23 b(b)s(e)f(a)h Fo(W)3272 3088 y Fw(\003)3311 3125 y Fu(-dynamical)0 3245 y(system.)0 3472 y Fj(De\014nition)38 b(2.15)49 b Fg(L)-5 b(et)36 b Fo(\014)f(>)29 b Fu(0)p Fg(.)47 b Fo(!)32 b Fn(2)e Fp(M)1555 3436 y Fk(+)p Fl(;)p Fk(1)1555 3497 y Fw(\003)1704 3472 y Fg(is)35 b(c)-5 b(al)5 b(le)-5 b(d)35 b(a)g Fu(\()p Fo(\034)6 b(;)17 b(\014)6 b Fu(\))p Fg(-KMS)35 b(state)h(if)f(for)h(any)f Fo(A;)17 b(B)34 b Fn(2)c Fp(M)0 3592 y Fg(ther)-5 b(e)32 b(exists)g(a)g(function)g Fo(F)1030 3607 y Fl(A;B)1164 3592 y Fu(\()p Fo(z)t Fu(\))p Fg(,)h(analytic)f(in)g(the)g(strip)h Fn(f)p Fo(z)f Fu(:)c(0)f Fo(<)h Fu(Im)p Fo(z)33 b(<)27 b(\014)6 b Fn(g)p Fg(,)32 b(c)-5 b(ontinuous)32 b(on)g(its)0 3713 y(closur)-5 b(e,)34 b(and)h(satisfying)f(the)h(KMS)g(b)-5 b(oundary)35 b(c)-5 b(onditions)33 b(for)i Fo(t)28 b Fn(2)g Fm(R)p Fg(:)1535 3923 y Fo(F)1598 3938 y Fl(A;B)1731 3923 y Fu(\()p Fo(t)p Fu(\))g(=)g Fo(!)t Fu(\()p Fo(A\034)2203 3882 y Fl(t)2232 3923 y Fu(\()p Fo(B)5 b Fu(\)\))p Fo(;)1327 4139 y(F)1390 4154 y Fl(A;B)1524 4139 y Fu(\()p Fo(t)22 b Fu(+)g(i)p Fo(\014)6 b Fu(\))27 b(=)h Fo(!)t Fu(\()p Fo(\034)2130 4098 y Fl(t)2159 4139 y Fu(\()p Fo(B)5 b Fu(\))p Fo(A)p Fu(\))p Fo(:)0 4365 y Fj(Theorem)75 b(2.16)49 b Fg(L)-5 b(et)36 b Fo(!)i Fg(b)-5 b(e)34 b(a)h Fu(\()p Fo(\034)6 b(;)17 b(\014)6 b Fu(\))p Fg(-KMS)34 b(state)h(and)g Fo(\014)e(>)27 b Fu(0)p Fg(.)45 b(Then)41 4516 y Fu(\(1\))j Fo(!)39 b Fg(is)34 b Fo(\034)11 b Fg(-invariant.)41 4678 y Fu(\(2\))48 b(s)252 4693 y Fl(!)331 4678 y Fu(=)28 b(z)478 4693 y Fl(!)528 4678 y Fg(.)45 b Fu(\()p Fg(In)34 b(p)-5 b(articular,)35 b Fo(!)j Fg(is)d(faithful)f(on)h Fu(z)1964 4693 y Fl(!)2015 4678 y Fp(M)p Fu(\))p Fg(.)41 4840 y Fu(\(3\))48 b Fg(If)35 b Fo(B)e Fn(2)28 b Fu(z)561 4855 y Fl(!)611 4840 y Fp(Z)p Fg(,)35 b(wher)-5 b(e)34 b Fp(Z)h Fg(is)g(the)g(c)-5 b(enter)34 b(of)h Fp(M)p Fg(,)f(then)h Fo(\034)2219 4804 y Fl(t)2249 4840 y Fu(\()p Fo(B)5 b Fu(\))28 b(=)f Fo(B)5 b Fg(.)41 5002 y Fu(\(4\))48 b Fg(L)-5 b(et)36 b Fo(\034)425 5017 y Fl(!)510 5002 y Fg(b)-5 b(e)35 b(the)g(dynamics)f(on)g Fu(z)1405 5017 y Fl(!)1456 5002 y Fp(M)g Fg(gener)-5 b(ate)g(d)34 b(by)i Fo(!)t Fg(.)44 b(Then)1720 5220 y Fo(\034)1773 5179 y Fl(t)1803 5136 y Fc(\014)1803 5196 y(\014)1837 5260 y Fk(z)1868 5268 y Fb(!)1913 5260 y Fh(M)2017 5220 y Fu(=)27 b Fo(\034)2173 5179 y Fl(\014)s(t)2162 5245 y(!)2247 5220 y Fo(:)1841 5753 y Fu(13)p eop %%Page: 14 14 14 13 bop 0 407 a Fj(Theorem)75 b(2.17)49 b Fg(L)-5 b(et)29 b Fo(!)i Fg(b)-5 b(e)28 b(a)g(faithful)g(state)g(on)g Fp(M)g Fg(and)f Fo(\034)2249 422 y Fl(!)2328 407 y Fg(the)h(c)-5 b(orr)g(esp)g(onding)27 b(dynamics.)41 b(Then)0 527 y Fo(!)d Fg(is)d(a)f Fu(\()p Fo(\034)368 542 y Fl(!)419 527 y Fo(;)17 b Fu(1\))p Fg(-KMS)34 b(state.)146 753 y Fu(Let)28 b(\()p Fp(M)p Fo(;)17 b Fn(H)q Fo(;)g(J)n(;)g Fn(H)813 717 y Fk(+)871 753 y Fu(\))28 b(b)s(e)f(a)h(standard)g(form.) 41 b(W)-8 b(e)28 b(sa)m(y)h(that)e(\012)h(is)g(a)g(standard)g(\()p Fo(\034)6 b(;)17 b(\014)6 b Fu(\)-KMS)26 b(v)m(ector)0 873 y(i\013)33 b(it)f(is)h(a)f(standard)h(v)m(ector)h(represen)m(tativ) m(e)i(of)c(a)g(\()p Fo(\034)6 b(;)17 b(\014)6 b Fu(\)-KMS)32 b(state.)146 993 y(Supp)s(ose)39 b(that)f Fo(L)g Fu(is)h(the)f (Liouvillean)h(of)e Fo(\034)11 b Fu(.)60 b(The)39 b(follo)m(wing)f (theorem)h(giv)m(es)g(a)f(criterium)h(for)0 1114 y(the)33 b(KMS)g(prop)s(ert)m(y)g(expressed)j(in)c(terms)i(of)e(Hilb)s(ert)h (spaces.)0 1339 y Fj(Theorem)75 b(2.18)49 b Fg(L)-5 b(et)36 b Fu(\012)28 b Fn(2)g(H)1206 1303 y Fk(+)1300 1339 y Fg(b)-5 b(e)34 b(a)h(unit)g(ve)-5 b(ctor.)45 b(Then)41 1491 y Fu(\(1\))j(\012)36 b Fg(is)e(a)h(standar)-5 b(d)34 b Fu(\()p Fo(\034)6 b(;)17 b(\014)6 b Fu(\))p Fg(-KMS)34 b(ve)-5 b(ctor)35 b(i\013)g Fp(M)p Fu(\012)27 b Fn(\032)h(D)s Fu(\(e)2302 1455 y Fw(\000)p Fl(\014)s(L=)p Fk(2)2523 1491 y Fu(\))35 b Fg(and)1367 1708 y Fu(e)1410 1667 y Fw(\000)p Fl(\014)s(L=)p Fk(2)1631 1708 y Fo(A)p Fu(\012)28 b(=)g Fo(J)9 b(A)2042 1667 y Fw(\003)2082 1708 y Fu(\012)p Fo(;)121 b(A)28 b Fn(2)g Fp(M)p Fo(:)41 1967 y Fu(\(2\))48 b Fg(If)35 b(in)f(addition)g Fu(\012)i Fg(is)e(cyclic)h(and)f Fu(\001)1569 1982 y Fk(\012)1659 1967 y Fg(is)h(the)g(c)-5 b(orr)g(esp)g(onding)33 b(mo)-5 b(dular)34 b(op)-5 b(er)g(ator,)35 b(then)1753 2184 y Fu(\001)1834 2199 y Fk(\012)1917 2184 y Fu(=)28 b(e)2064 2143 y Fw(\000)p Fl(\014)s(L)2214 2184 y Fo(:)0 2611 y Fs(2.13)136 b(Con)l(v)l(ergence)0 2796 y Fu(It)29 b(is)g(often)f(con)m(v)m(enien)m(t)k(to)c(reduce)i(the) f(study)h(of)e Fo(W)1979 2760 y Fw(\003)2018 2796 y Fu(-dynamics)i(and) f(normal)f(states)i(to)e(the)h(study)0 2916 y(of)k(corresp)s(onding)i (Liouvilleans)g(and)f(standard)g(v)m(ector)g(represen)m(tativ)m(es.)51 b(In)33 b(this)i(subsection)g(w)m(e)0 3037 y(apply)i(this)g(p)s(oin)m (t)f(of)g(view)h(to)f(the)h(con)m(v)m(ergence)h(prop)s(erties)g(of)d Fo(W)2587 3001 y Fw(\003)2626 3037 y Fu(-dynamics,)k(in)m(v)-5 b(arian)m(t)37 b(states)0 3157 y(and)c(KMS)g(states.)0 3383 y Fj(Theorem)75 b(2.19)49 b Fg(Assume)35 b(that)h Fu(\()p Fp(M)p Fo(;)17 b Fn(H)q Fo(;)g(J)n(;)g Fn(H)1821 3346 y Fk(+)1879 3383 y Fu(\))34 b Fg(is)h(a)g Fo(W)2247 3346 y Fw(\003)2286 3383 y Fg(-algebr)-5 b(a)34 b(in)g(the)h(standar)-5 b(d)34 b(form.)41 3534 y Fu(\(1\))48 b Fg(Supp)-5 b(ose)25 b(that)h Fo(\034)808 3549 y Fl(n)880 3534 y Fg(is)f(a)g(se)-5 b(quenc)g(e)24 b(of)h Fo(W)1647 3498 y Fw(\003)1686 3534 y Fg(-dynamics)g(with)g(Liouvil)5 b(le)-5 b(ans)24 b Fo(L)2943 3549 y Fl(n)2990 3534 y Fg(,)j Fo(L)f Fg(is)f(a)g (self-adjoint)214 3655 y(op)-5 b(er)g(ator,)35 b(and)f Fo(L)883 3670 y Fl(n)958 3655 y Fn(!)28 b Fo(L)35 b Fg(in)f(the)h(str) -5 b(ong)35 b(r)-5 b(esolvent)34 b(sense.)44 b(Then)1597 3872 y Fo(\034)1650 3831 y Fl(t)1680 3872 y Fu(\()p Fo(A)p Fu(\))28 b(:=)g(e)2031 3831 y Fk(i)p Fl(tL)2128 3872 y Fo(A)p Fu(e)2244 3831 y Fw(\000)p Fk(i)p Fl(tL)214 4090 y Fg(is)35 b(a)g Fo(W)510 4053 y Fw(\003)549 4090 y Fg(-dynamics)f(on)g Fp(M)h Fg(and)f Fo(L)h Fg(is)g(its)g(Liouvil)5 b(le)-5 b(an.)41 4251 y Fu(\(2\))48 b Fg(Assume)33 b(in)f(addition)g (that)h Fo(!)1330 4266 y Fl(n)1404 4251 y Fn(2)28 b Fp(M)1603 4215 y Fk(+)1603 4276 y Fw(\003)1694 4251 y Fg(ar)-5 b(e)32 b Fo(\034)1899 4266 y Fl(n)1947 4251 y Fg(-invariant)f(and)h Fu(\012)2658 4266 y Fl(n)2738 4251 y Fg(ar)-5 b(e)32 b(their)h(standar)-5 b(d)32 b(ve)-5 b(ctor)214 4371 y(r)g(epr)g (esentatives.)42 b(Supp)-5 b(ose)25 b(also)i(that)g Fu(w)q Fn(\000)17 b Fu(lim)1958 4386 y Fl(n)2022 4371 y Fu(\012)2092 4386 y Fl(n)2167 4371 y Fu(=)27 b(\012)p Fg(.)43 b(Then)25 b Fu(\012)k Fn(2)f(H)2936 4335 y Fk(+)3021 4371 y Fg(and)e(the)h (functional)214 4492 y Fo(!)275 4507 y Fk(\012)365 4492 y Fg(is)35 b Fo(\034)11 b Fg(-invariant.)41 4653 y Fu(\(3\))48 b Fg(Assume)36 b(in)g(addition)f(that)h Fo(!)1343 4668 y Fl(n)1426 4653 y Fg(ar)-5 b(e)35 b Fu(\()p Fo(\034)1672 4668 y Fl(n)1719 4653 y Fo(;)17 b(\014)6 b Fu(\))p Fg(-KMS)35 b(states)h(and)g(that)g Fu(\012)30 b Fn(6)p Fu(=)f(0)p Fg(.)47 b(Then)35 b Fo(!)3462 4669 y Fk(\012)p Fl(=)p Fw(k)p Fk(\012)p Fw(k)3710 4653 y Fg(is)214 4774 y(a)g Fu(\()p Fo(\034)6 b(;)17 b(\014)6 b Fu(\))p Fg(-KMS)34 b(state.)0 5030 y Fj(Pro)s(of.)65 b Fu(\(1\))32 b(Let)h Fo(A)28 b Fn(2)g Fp(M)p Fu(.)43 b(W)-8 b(e)33 b(ha)m(v)m(e)h(s)p Fn(\000)17 b Fu(lim)1731 5045 y Fl(n)p Fw(!1)1935 5030 y Fu(e)1978 4994 y Fw(\006)p Fk(i)p Fl(tL)2126 5002 y Fb(n)2201 5030 y Fu(=)28 b(e)2348 4994 y Fw(\006)p Fk(i)p Fl(tL)2500 5030 y Fu(,)33 b(hence)1079 5248 y(s)p Fn(\000)42 b Fu(lim)1212 5308 y Fl(n)p Fw(!1)1412 5248 y Fu(e)1455 5207 y Fk(i)p Fl(tL)1548 5215 y Fb(n)1595 5248 y Fo(A)p Fu(e)1711 5207 y Fw(\000)p Fk(i)p Fl(tL)1859 5215 y Fb(n)1934 5248 y Fu(=)28 b(e)2081 5207 y Fk(i)p Fl(tL)2178 5248 y Fo(A)p Fu(e)2294 5207 y Fw(\000)p Fk(i)p Fl(tL)2475 5248 y Fn(2)g Fp(M)p Fo(:)1841 5753 y Fu(14)p eop %%Page: 15 15 15 14 bop 0 407 a Fu(Therefore)34 b Fo(\034)44 b Fu(is)33 b(a)f Fo(W)813 371 y Fw(\003)852 407 y Fu(-dynamics.)146 527 y(Since)27 b Fn(H)479 491 y Fk(+)563 527 y Fu(is)f(closed)g(and)f (e)1159 491 y Fk(i)p Fl(tL)1252 499 y Fb(n)1325 527 y Fu(preserv)m(e)i Fn(H)1782 491 y Fk(+)1841 527 y Fu(,)g(e)1938 491 y Fk(i)p Fl(tL)2061 527 y Fu(preserv)m(es)h Fn(H)2557 491 y Fk(+)2616 527 y Fu(.)41 b(Hence)27 b Fo(L)e Fu(is)h(the)f (Liouvillean)0 648 y(of)32 b Fo(\034)11 b Fu(.)146 768 y(\(2\))34 b(Since)i Fn(H)647 732 y Fk(+)740 768 y Fu(is)f(w)m(eakly)h (closed,)g(\012)31 b Fn(2)g(H)1764 732 y Fk(+)1823 768 y Fu(.)49 b(Moreo)m(v)m(er,)36 b(since)g(\012)2667 783 y Fl(n)2745 768 y Fn(2)30 b(D)s Fu(\()p Fo(L)3025 783 y Fl(n)3072 768 y Fu(\))35 b(and)f Fo(L)3402 783 y Fl(n)3449 768 y Fu(\012)3519 783 y Fl(n)3597 768 y Fu(=)d(0,)0 888 y(b)m(y)i(Prop)s(osition)g(A.6,)g(\012)28 b Fn(2)g(D)s Fu(\()p Fo(L)p Fu(\))33 b(and)f Fo(L)p Fu(\012)d(=)e(0.)146 1009 y(\(3\))32 b(Let)h Fo(A)28 b Fn(2)g Fp(M)p Fu(.)43 b(\012)918 1024 y Fl(n)998 1009 y Fu(are)33 b(\()p Fo(\034)1241 1024 y Fl(n)1288 1009 y Fo(;)17 b(\014)6 b Fu(\)-KMS)32 b(v)m(ectors,)i(hence)1282 1208 y(exp)q(\()p Fn(\000)p Fo(\014)6 b(L)1673 1223 y Fl(n)1720 1208 y Fo(=)p Fu(2\))p Fo(A)p Fu(\012)1999 1223 y Fl(n)2074 1208 y Fu(=)27 b Fo(J)9 b(A)2313 1167 y Fw(\003)2353 1208 y Fu(\012)2423 1223 y Fl(n)2471 1208 y Fo(:)0 1407 y Fu(Since)25 b(exp)q(\()p Fn(\000)p Fo(\014)6 b(L)637 1422 y Fl(n)685 1407 y Fo(=)p Fu(2\))27 b Fn(!)g Fu(exp)q(\()p Fn(\000)p Fo(\014)6 b(L=)p Fu(2\))24 b(in)g(the)h(strong)f(resolv)m(en)m(t)i(sense,)h Fo(J)9 b(A)2888 1371 y Fw(\003)2928 1407 y Fu(\012)2998 1422 y Fl(n)3073 1407 y Fn(!)28 b Fo(J)9 b(A)3337 1371 y Fw(\003)3376 1407 y Fu(\012)25 b(w)m(eakly)-8 b(,)0 1527 y(and)33 b Fo(A)p Fu(\012)333 1542 y Fl(n)408 1527 y Fn(!)27 b Fo(A)p Fu(\012)33 b(w)m(eakly)-8 b(,)35 b(it)d(follo)m(ws)i (from)e(Prop)s(osition)h(A.6)f(that)h Fo(A)p Fu(\012)28 b Fn(2)g(D)s Fu(\(e)3045 1491 y Fw(\000)p Fl(\014)s(L=)p Fk(2)3265 1527 y Fu(\))33 b(and)1484 1727 y(e)1527 1685 y Fw(\000)p Fl(\014)s(L=)p Fk(2)1748 1727 y Fo(A)p Fu(\012)28 b(=)g Fo(J)9 b(A)2159 1685 y Fw(\003)2198 1727 y Fu(\012)p Fo(:)1284 b Fu(\(2.5\))0 1926 y(Hence)34 b(\012)p Fo(=)p Fn(k)p Fu(\012)p Fn(k)f Fu(is)g(a)f(\()p Fo(\034)6 b(;)17 b(\014)6 b Fu(\)-KMS)32 b(v)m(ector.)44 b Fe(2)0 2255 y Fs(2.14)136 b(Analytic)45 b(elemen)l(ts)0 2440 y Fu(Let)37 b(\()p Fp(M)p Fo(;)17 b(\034)11 b Fu(\))37 b(b)s(e)g(a)g Fo(W)823 2404 y Fw(\003)862 2440 y Fu(-dynamical)h(system.)59 b(An)37 b(elemen)m(t)i Fo(A)c Fn(2)h Fp(M)h Fu(is)g(called)h Fo(\034)11 b Fu(-analytic)38 b(if)f(there)0 2560 y(exists)d(a)f(strip)g Fo(I)8 b Fu(\()p Fo(r)s Fu(\))27 b(=)g Fn(f)p Fo(z)33 b Fu(:)27 b Fn(j)p Fu(Im)q Fo(z)t Fn(j)h Fo(<)f(r)s Fn(g)32 b Fu(and)h(a)f(function)h Fo(f)39 b Fu(:)27 b Fo(I)8 b Fu(\()p Fo(r)s Fu(\))28 b Fn(7!)f Fp(M)32 b Fu(suc)m(h)i(that:)0 2681 y(\(1\))e Fo(f)11 b Fu(\()p Fo(t)p Fu(\))28 b(=)f Fo(\034)511 2645 y Fl(t)541 2681 y Fu(\()p Fo(A)p Fu(\))33 b(for)f Fo(t)c Fn(2)g Fm(R)p Fu(;)0 2823 y(\(2\))k Fo(I)8 b Fu(\()p Fo(r)s Fu(\))27 b Fn(3)h Fo(z)33 b Fn(7!)27 b Fo(\036)p Fu(\()p Fo(f)11 b Fu(\()p Fo(z)t Fu(\)\))33 b(is)g(analytic)g(for)f(all)g Fo(\036)c Fn(2)g Fp(M)2046 2838 y Fw(\003)2085 2823 y Fu(.)146 3086 y(Under)40 b(these)h (conditions)f(w)m(e)g(write)g Fo(f)11 b Fu(\()p Fo(z)t Fu(\))39 b(=)g Fo(\034)1975 3050 y Fl(z)2016 3086 y Fu(\()p Fo(A)p Fu(\).)63 b(A)39 b(standard)h(argumen)m(t)g(based)g(on)f(the)0 3206 y(uniform)33 b(b)s(oundedness)i(theorem)e(sho)m(ws)h(that)f Fo(f)11 b Fu(\()p Fo(z)t Fu(\))32 b(is)i(actually)f(analytic)g(in)g (the)g(norm)f(of)h Fp(M)p Fu(.)146 3327 y(If)g Fo(r)d Fu(=)e Fn(1)p Fu(,)k(then)h(w)m(e)h(sa)m(y)f(that)g Fo(A)f Fu(is)h Fo(\034)11 b Fu(-en)m(tire.)146 3447 y(F)-8 b(or)32 b Fo(A)c Fn(2)g Fp(M)k Fu(and)h Fo(n)28 b Fn(2)g Fm(N)k Fu(let)1291 3685 y Fo(A)1364 3700 y Fl(n)1438 3685 y Fu(=)1542 3575 y Fc(\020)1612 3618 y Fo(n)p 1611 3663 59 4 v 1611 3754 a(\031)1680 3575 y Fc(\021)1750 3570 y Ff(1)p 1750 3582 31 3 v 1750 3623 a(2)1811 3550 y Fc(Z)1866 3775 y Fa(R)1935 3685 y Fu(e)1978 3644 y Fw(\000)p Fl(nt)2101 3621 y Ff(2)2140 3685 y Fo(\034)2193 3644 y Fl(t)2223 3685 y Fu(\()p Fo(A)p Fu(\)d)p Fo(t:)0 3932 y Fj(Theorem)75 b(2.20)49 b Fo(A)833 3947 y Fl(n)915 3932 y Fg(is)35 b Fo(\034)11 b Fg(-entir)-5 b(e)35 b(and)f Fo(A)1648 3947 y Fl(n)1723 3932 y Fn(\045)27 b Fo(A)35 b Fg(in)g(the)g Fo(\033)t Fg(-str)-5 b(ong)34 b(top)-5 b(olo)g(gy.)45 b(Thus)34 b(the)h Fo(\034)11 b Fg(-entir)-5 b(e)0 4052 y(elements)34 b(form)g(a)h Fo(\033)t Fg(-str)-5 b(ongly)35 b(dense)f(subsp)-5 b(ac)g(e)34 b(of)g Fp(M)p Fg(.)45 b(This)34 b(subsp)-5 b(ac)g(e)34 b(is)h(denote)-5 b(d)34 b(by)h Fp(M)3442 4067 y Fl(\034)3484 4052 y Fg(.)146 4238 y Fu(F)-8 b(or)32 b(additional)h(discussion)i(of)d(analytic)h (elemen)m(ts)i(w)m(e)e(refer)g(the)g(reader)g(to)g([BR1)o(].)0 4567 y Ft(3)161 b(The)53 b(p)t(erturbation)f(theory)h(of)h Fi(W)2372 4515 y Fn(\003)2425 4567 y Ft(-dynamics)0 4786 y Fu(In)35 b(this)g(section,)h(giv)m(en)g(a)e Fo(W)1117 4750 y Fw(\003)1156 4786 y Fu(-dynamics)i Fo(\034)46 b Fu(and)34 b(a)h(p)s(erturbation)f Fo(Q)p Fu(,)h(w)m(e)h(construct)g (a)e(p)s(erturb)s(ed)0 4907 y Fo(W)106 4871 y Fw(\003)145 4907 y Fu(-dynamics)f Fo(\034)650 4922 y Fl(Q)710 4907 y Fu(.)43 b(W)-8 b(e)31 b(also)g(construct)i(the)e(so-called)h (Araki-Dyson)g(expansionals)h(E)3331 4871 y Fl(\034)3331 4932 y(Q)3391 4907 y Fu(\()p Fo(t)p Fu(\))e(whic)m(h)0 5027 y(in)m(tert)m(wine)45 b(these)f(t)m(w)m(o)f(dynamics.)76 b(W)-8 b(e)43 b(describ)s(e)i(these)f(ob)5 b(jects)44 b(in)f(three)g(cases:)66 b(for)42 b(analytic)0 5147 y(p)s (erturbations,)33 b(b)s(ounded)h(p)s(erturbations,)f(and)f(for)g(a)h (large)f(class)i(of)e(un)m(b)s(ounded)i(p)s(erturbations.)0 5268 y(The)g(constructions)g(in)f(the)g(\014rst)g(t)m(w)m(o)g(cases)h (are)e(w)m(ell)i(kno)m(wn,)g(see)g([Ar6,)e(BR2].)1841 5753 y(15)p eop %%Page: 16 16 16 15 bop 0 407 a Fs(3.1)135 b(Bounded)44 b(p)t(erturbations)0 592 y Fu(Let)33 b(\()p Fp(M)p Fo(;)17 b(\034)11 b Fu(\))32 b(b)s(e)h(a)f Fo(W)805 555 y Fw(\003)844 592 y Fu(-dynamical)i(system)h (and)d Fo(Q)h Fu(a)g(self-adjoin)m(t)g(elemen)m(t)h(of)e Fp(M)p Fu(.)44 b(The)33 b(follo)m(wing)0 712 y(form)m(ula)g(de\014nes)h (the)f Fo(W)952 676 y Fw(\003)991 712 y Fu(-dynamics)h Fo(\034)1497 727 y Fl(Q)1589 712 y Fu(on)f Fp(M)p Fu(:)429 953 y Fo(\034)482 912 y Fl(t)471 977 y(Q)531 953 y Fu(\()p Fo(A)p Fu(\))28 b(=)811 858 y Fc(X)817 1069 y Fl(n)p Fw(\025)p Fk(0)972 953 y Fu(i)999 912 y Fl(n)1063 817 y Fc(Z)1118 1043 y Fk(0)p Fw(\024)p Fl(t)1233 1051 y Fb(n)1276 1043 y Fw(\024)p Fl(:::)o(t)1415 1052 y Ff(1)1450 1043 y Fw(\024)p Fl(t)1534 953 y Fu([)p Fo(\034)1614 912 y Fl(t)1639 920 y Fb(n)1687 953 y Fu(\()p Fo(Q)p Fu(\))p Fo(;)17 b Fu([)p Fn(\001)g(\001)g(\001)31 b Fo(;)17 b Fu([)p Fo(\034)2184 912 y Fl(t)2209 921 y Ff(1)2249 953 y Fu(\()p Fo(Q)p Fu(\))p Fo(;)g(\034)2499 912 y Fl(t)2529 953 y Fu(\()p Fo(A)p Fu(\)])g Fn(\001)g(\001)g(\001)d Fu(]]d)p Fo(t)2997 968 y Fk(1)3054 953 y Fn(\001)j(\001)g(\001)e Fu(d)p Fo(t)3276 968 y Fl(n)3323 953 y Fo(:)229 b Fu(\(3.6\))146 1244 y(If)42 b Fo(\016)k Fu(is)d(the)f(generator)g(of)g Fo(\034)11 b Fu(,)45 b(then)e(the)f(generator)g(of)g Fo(\034)2342 1208 y Fl(t)2331 1269 y(Q)2433 1244 y Fu(has)g(the)h(same) g(domain)f(as)g Fo(\016)k Fu(and)0 1365 y(equals)1389 1485 y Fo(\016)1432 1500 y Fl(Q)1492 1485 y Fu(\()p Fo(A)p Fu(\))27 b(=)h Fo(\016)t Fu(\()p Fo(A)p Fu(\))22 b(+)g(i[)p Fo(Q;)17 b(A)p Fu(])p Fo(:)146 1646 y Fu(Let)33 b(E)387 1610 y Fl(\034)387 1671 y(Q)447 1646 y Fu(\()p Fo(t)p Fu(\))g(b)s(e)g(a)f(one-parameter)h(family)g(of)f(elemen)m(ts)j(of)d Fp(M)g Fu(giv)m(en)i(b)m(y)765 1897 y(E)831 1856 y Fl(\034)831 1922 y(Q)891 1897 y Fu(\()p Fo(t)p Fu(\))28 b(=)1134 1803 y Fc(X)1139 2013 y Fl(n)p Fw(\025)p Fk(0)1294 1897 y Fu(i)1321 1856 y Fl(n)1385 1762 y Fc(Z)1440 1987 y Fk(0)p Fw(\024)p Fl(t)1555 1995 y Fb(n)1598 1987 y Fw(\024)p Fl(:::)o(t)1737 1996 y Ff(1)1772 1987 y Fw(\024)p Fl(t)1873 1897 y Fo(\034)1926 1856 y Fl(t)1951 1864 y Fb(n)1999 1897 y Fu(\()p Fo(Q)p Fu(\))17 b Fn(\001)g(\001)g(\001)d Fo(\034)2354 1856 y Fl(t)2379 1865 y Ff(1)2419 1897 y Fu(\()p Fo(Q)p Fu(\)d)p Fo(t)2661 1912 y Fk(1)2718 1897 y Fn(\001)j(\001)g(\001)e Fu(d)p Fo(t)2940 1912 y Fl(n)2987 1897 y Fo(:)3579 1923 y Fu(\(3.7\))0 2186 y(W)-8 b(e)33 b(will)g(call)g(E)597 2150 y Fl(\034)597 2212 y(Q)657 2186 y Fu(\()p Fo(t)p Fu(\))g(the)g(Araki-Dyson)g(expansionals.)45 b(Whenev)m(er)35 b(there)e(is)g(no)g(danger)g(of)f(confu-)0 2307 y(sion)h(w)m(e)h(will)f(write)g(E)844 2322 y Fl(Q)904 2307 y Fu(\()p Fo(t)p Fu(\))g(for)f(E)1263 2271 y Fl(\034)1263 2332 y(Q)1323 2307 y Fu(\()p Fo(t)p Fu(\).)146 2427 y(W)-8 b(e)32 b(remark)h(that)f(in)m(tegrals)g(in)g(\(3.6\))f(and)h(\(3.7\))g (con)m(v)m(erge)h(in)f Fo(\033)t Fu(-w)m(eak)h(top)s(ology)e(and)h (de\014ne)h(a)0 2548 y(norm-con)m(v)m(ergen)m(t)h(series)h(of)d(b)s (ounded)h(op)s(erators.)146 2668 y(The)39 b(expansions)h(\(3.6\))d(and) h(\(3.7\))f(pla)m(y)m(ed)j(an)e(imp)s(ortan)m(t)g(role)f(in)i(the)f(w)m (orks)h(of)e(Sc)m(h)m(winger,)0 2788 y(T)-8 b(omonaga)36 b(and)h(Dyson)g(on)f(QED.)g(The)i(op)s(erators)e(E)2101 2752 y Fl(\034)2101 2814 y(Q)2161 2788 y Fu(\()p Fo(t)p Fu(\))h(are)f(closely)j(related)e(to)f(the)h(so-called)0 2909 y(Connes)d(co)s(cycles)g([Co)q(].)146 3029 y(Let)f(us)g(list)g (some)h(prop)s(erties)f(of)f(Araki-Dyson)h(expansionals:)0 3223 y Fj(Theorem)75 b(3.1)49 b Fg(L)-5 b(et)36 b Fo(t;)17 b(t)987 3238 y Fk(1)1026 3223 y Fo(;)g(t)1105 3238 y Fk(2)1172 3223 y Fn(2)28 b Fm(R)p Fg(.)45 b(Then)41 3374 y Fu(\(1\))j(E)280 3389 y Fl(Q)341 3374 y Fu(\()p Fo(t)p Fu(\))34 b Fg(ar)-5 b(e)35 b(unitary)h(elements)e(of)g Fp(M)p Fg(;)41 3532 y Fu(\(2\))48 b Fo(\034)267 3495 y Fl(t)256 3557 y(Q)316 3532 y Fu(\()p Fo(A)p Fu(\))28 b(=)g(E)663 3495 y Fl(\034)663 3557 y(Q)723 3532 y Fu(\()p Fo(t)p Fu(\))p Fo(\034)887 3495 y Fl(t)917 3532 y Fu(\()p Fo(A)p Fu(\)E)1132 3495 y Fl(\034)1132 3557 y(Q)1192 3532 y Fu(\()p Fo(t)p Fu(\))1303 3495 y Fw(\000)p Fk(1)1397 3532 y Fu(;)41 3689 y(\(3\))48 b(E)280 3704 y Fl(Q)341 3689 y Fu(\()p Fo(t)p Fu(\))452 3653 y Fw(\000)p Fk(1)574 3689 y Fu(=)27 b(E)743 3704 y Fl(Q)803 3689 y Fu(\()p Fo(t)p Fu(\))914 3653 y Fw(\003)981 3689 y Fu(=)h Fo(\034)1138 3653 y Fl(t)1168 3689 y Fu(\(E)1272 3704 y Fl(Q)1332 3689 y Fu(\()p Fn(\000)p Fo(t)p Fu(\)\))p Fg(;)41 3847 y Fu(\(4\))48 b(E)280 3862 y Fl(Q)341 3847 y Fu(\()p Fo(t)414 3862 y Fk(1)475 3847 y Fu(+)22 b Fo(t)608 3862 y Fk(2)648 3847 y Fu(\))28 b(=)f(E)883 3862 y Fl(Q)943 3847 y Fu(\()p Fo(t)1016 3862 y Fk(1)1056 3847 y Fu(\))p Fo(\034)1147 3811 y Fl(t)1172 3820 y Ff(1)1211 3847 y Fu(\(E)1315 3862 y Fl(Q)1375 3847 y Fu(\()p Fo(t)1448 3862 y Fk(2)1488 3847 y Fu(\)\))p Fg(;)0 3998 y(Assume)32 b(in)h(addition)e(that)i Fp(M)f Fg(is)g(a)g(c)-5 b(oncr)g(ete)32 b Fo(W)1861 3962 y Fw(\003)1900 3998 y Fg(-algebr)-5 b(a)32 b(in)g Fn(B)s Fu(\()p Fn(H)q Fu(\))h Fg(and)f(that)h Fo(L)f Fg(is)h(a)f(self-adjoint)0 4119 y(op)-5 b(er)g(ator)35 b(on)f Fn(H)i Fg(such)f(that)g Fo(\034)1117 4082 y Fl(t)1147 4119 y Fu(\()p Fo(A)p Fu(\))28 b(=)f(e)1470 4082 y Fk(i)p Fl(tL)1568 4119 y Fo(A)p Fu(e)1684 4082 y Fw(\000)p Fk(i)p Fl(tL)1871 4119 y Fg(for)35 b Fo(A)28 b Fn(2)g Fp(M)p Fg(.)44 b(Then)41 4270 y Fu(\(5\))k Fo(\034)267 4234 y Fl(t)256 4295 y(Q)316 4270 y Fu(\()p Fo(A)p Fu(\))28 b(=)g(e)640 4234 y Fk(i)p Fl(t)p Fk(\()p Fl(L)p Fk(+)p Fl(Q)p Fk(\))903 4270 y Fo(A)p Fu(e)1019 4234 y Fw(\000)p Fk(i)p Fl(t)p Fk(\()p Fl(L)p Fk(+)p Fl(Q)p Fk(\))1371 4270 y Fg(for)35 b Fo(A)28 b Fn(2)g Fp(M)p Fg(;)41 4440 y Fu(\(6\))48 b(E)280 4455 y Fl(Q)341 4440 y Fu(\()p Fo(t)p Fu(\))27 b(=)h(e)626 4403 y Fk(i)p Fl(t)p Fk(\()p Fl(L)p Fk(+)p Fl(Q)p Fk(\))889 4440 y Fu(e)932 4403 y Fw(\000)p Fk(i)p Fl(tL)1084 4440 y Fg(.)0 4755 y Fs(3.2)135 b(Analytic)46 b(p)t(erturbations)0 4940 y Fu(In)33 b(this)g(subsection) i(w)m(e)e(assume)h(that)f Fo(Q)g Fu(is)g Fo(\034)11 b Fu(-en)m(tire.)44 b(Then)34 b Fo(\034)2379 4955 y Fl(Q)2472 4940 y Fu(extends)g(to)e Fm(C)h Fu(b)m(y)g(the)g(form)m(ula)204 5181 y Fo(\034)257 5140 y Fl(z)246 5205 y(Q)306 5181 y Fu(\()p Fo(A)p Fu(\))27 b(=)586 5086 y Fc(X)591 5296 y Fl(n)p Fw(\025)p Fk(0)730 5181 y Fu(\(i)p Fo(z)t Fu(\))882 5140 y Fl(n)946 5045 y Fc(Z)1002 5271 y Fk(0)p Fw(\024)p Fl(s)1125 5279 y Fb(n)1167 5271 y Fw(\024)p Fl(:::)o(s)1314 5280 y Ff(1)1348 5271 y Fw(\024)p Fk(1)1442 5181 y Fu([)p Fo(\034)1522 5140 y Fl(s)1555 5148 y Fb(n)1598 5140 y Fl(z)1638 5181 y Fu(\()p Fo(Q)p Fu(\))p Fo(;)17 b Fu([)p Fn(\001)g(\001)g(\001)31 b Fo(;)17 b Fu([)p Fo(\034)2135 5140 y Fl(s)2168 5149 y Ff(1)2202 5140 y Fl(z)2242 5181 y Fu(\()p Fo(Q)p Fu(\))p Fo(;)g(\034)2492 5140 y Fl(z)2532 5181 y Fu(\()p Fo(A)p Fu(\)])g Fn(\001)g(\001)g(\001)e Fu(]]d)p Fo(s)3012 5196 y Fk(1)3068 5181 y Fn(\001)i(\001)g(\001)e Fu(d)p Fo(s)3301 5196 y Fl(n)3348 5181 y Fo(;)204 b Fu(\(3.8\))1841 5753 y(16)p eop %%Page: 17 17 17 16 bop 0 407 a Fu(v)-5 b(alid)33 b(for)f Fo(A)c Fn(2)g Fp(M)685 422 y Fl(\034)727 407 y Fu(.)44 b(Th)m(us)34 b Fp(M)1150 422 y Fl(\034)1220 407 y Fu(=)28 b Fp(M)1429 422 y Fl(\034)1460 433 y Fb(Q)1516 407 y Fu(.)146 527 y(F)-8 b(or)32 b Fo(\034)11 b Fu(-analytic)33 b Fo(Q)p Fu(,)g(the)g(Araki-Dyson)g(expansionals)i(can)d(b)s(e)h(de\014ned)h (for)e(all)h(complex)h Fo(z)j Fu(b)m(y)638 790 y(E)704 749 y Fl(\034)704 815 y(Q)764 790 y Fu(\()p Fo(z)t Fu(\))28 b(=)1021 695 y Fc(X)1026 906 y Fl(n)p Fw(\025)p Fk(0)1165 790 y Fu(\(i)p Fo(z)t Fu(\))1317 749 y Fl(n)1381 654 y Fc(Z)1436 880 y Fk(0)p Fw(\024)p Fl(s)1559 888 y Fb(n)1602 880 y Fw(\024)p Fl(:::)n(s)1748 889 y Ff(1)1783 880 y Fw(\024)p Fk(1)1894 790 y Fo(\034)1947 749 y Fl(s)1980 757 y Fb(n)2022 749 y Fl(z)2062 790 y Fu(\()p Fo(Q)p Fu(\))17 b Fn(\001)g(\001)g(\001)e Fo(\034)2418 749 y Fl(s)2451 758 y Ff(1)2485 749 y Fl(z)2525 790 y Fu(\()p Fo(Q)p Fu(\)d)p Fo(s)2778 805 y Fk(1)2834 790 y Fn(\001)i(\001)g(\001)e Fu(d)p Fo(s)3067 805 y Fl(n)3114 790 y Fo(:)438 b Fu(\(3.9\))0 1110 y(The)39 b(series)h(\(3.8\))e(and)g(\(3.9\))g(con)m(v)m(erge)i(in) f(norm)f(uniformly)h(for)f Fo(z)43 b Fu(in)38 b(compact)h(sets)h(and)e (de\014ne)0 1230 y(analytic)33 b(functions)h(with)f(v)-5 b(alues)33 b(in)g Fp(M)p Fu(.)0 1458 y Fj(Theorem)75 b(3.2)49 b Fg(L)-5 b(et)36 b Fo(z)t(;)17 b(z)1011 1473 y Fk(1)1051 1458 y Fo(;)g(z)1140 1473 y Fk(2)1207 1458 y Fn(2)28 b Fm(C)p Fg(.)45 b(Then)41 1610 y Fu(\(1\))j(E)280 1625 y Fl(Q)341 1610 y Fu(\()p Fo(z)t Fu(\))28 b Fn(2)g Fp(M)693 1625 y Fl(\034)736 1610 y Fg(;)41 1772 y Fu(\(2\))48 b Fo(\034)267 1736 y Fl(z)256 1797 y(Q)316 1772 y Fu(\()p Fo(A)p Fu(\))28 b(=)g(E)663 1736 y Fl(\034)663 1797 y(Q)723 1772 y Fu(\()p Fo(z)t Fu(\))p Fo(\034)901 1736 y Fl(z)942 1772 y Fu(\()p Fo(A)p Fu(\)E)1157 1736 y Fl(\034)1157 1797 y(Q)1217 1772 y Fu(\()p Fo(z)t Fu(\))1342 1736 y Fw(\000)p Fk(1)1437 1772 y Fg(;)41 1934 y Fu(\(3\))48 b(E)280 1949 y Fl(Q)341 1934 y Fu(\()p Fo(z)t Fu(\))466 1898 y Fw(\000)p Fk(1)588 1934 y Fu(=)28 b(E)758 1949 y Fl(Q)818 1934 y Fu(\()p 856 1879 50 4 v Fo(z)t Fu(\))943 1898 y Fw(\003)1010 1934 y Fu(=)g Fo(\034)1167 1898 y Fl(z)1207 1934 y Fu(\(E)1311 1949 y Fl(Q)1371 1934 y Fu(\()p Fn(\000)p Fo(z)t Fu(\)\);)41 2096 y(\(4\))48 b(E)280 2111 y Fl(Q)341 2096 y Fu(\()p Fo(z)424 2111 y Fk(1)485 2096 y Fu(+)22 b Fo(z)628 2111 y Fk(2)668 2096 y Fu(\))28 b(=)f(E)903 2111 y Fl(Q)963 2096 y Fu(\()p Fo(z)1046 2111 y Fk(1)1086 2096 y Fu(\))p Fo(\034)1177 2059 y Fl(z)1210 2068 y Ff(1)1249 2096 y Fu(\(E)1353 2111 y Fl(Q)1413 2096 y Fu(\()p Fo(z)1496 2111 y Fk(2)1536 2096 y Fu(\)\))p Fg(;)0 2247 y(Assume)32 b(in)h(addition)e(that)i Fp(M)f Fg(is)g(a)g(c)-5 b(oncr)g(ete)32 b Fo(W)1861 2211 y Fw(\003)1900 2247 y Fg(-algebr)-5 b(a)32 b(in)g Fn(B)s Fu(\()p Fn(H)q Fu(\))h Fg(and)f(that)h Fo(L)f Fg(is)h(a)f(self-adjoint) 0 2367 y(op)-5 b(er)g(ator)35 b(on)f Fn(H)i Fg(such)f(that)g Fo(\034)1117 2331 y Fl(t)1147 2367 y Fu(\()p Fo(A)p Fu(\))28 b(=)f(e)1470 2331 y Fk(i)p Fl(tL)1568 2367 y Fo(A)p Fu(e)1684 2331 y Fw(\000)p Fk(i)p Fl(tL)1871 2367 y Fg(for)35 b Fo(A)28 b Fn(2)g Fp(M)p Fg(.)44 b(Then)41 2519 y Fu(\(5\))k Fo(\034)267 2483 y Fl(z)256 2544 y(Q)316 2519 y Fu(\()p Fo(A)p Fu(\)e)508 2483 y Fk(i)p Fl(z)s Fk(\()p Fl(L)p Fk(+)p Fl(Q)p Fk(\))809 2519 y Fu(=)28 b(e)956 2483 y Fk(i)p Fl(z)s Fk(\()p Fl(L)p Fk(+)p Fl(Q)p Fk(\))1229 2519 y Fo(A)35 b Fg(for)f Fo(A)28 b Fn(2)g Fp(M)1792 2534 y Fl(\034)1835 2519 y Fg(;)41 2693 y Fu(\(6\))48 b(E)280 2708 y Fl(Q)341 2693 y Fu(\()p Fo(z)t Fu(\)e)509 2657 y Fk(i)p Fl(z)s(L)645 2693 y Fu(=)27 b(e)791 2657 y Fk(i)p Fl(z)s Fk(\()p Fl(L)p Fk(+)p Fl(Q)p Fk(\))1064 2693 y Fg(.)0 3013 y Fs(3.3)135 b(Un)l(b)t(ounded)44 b(p)t(erturbations)0 3198 y Fu(In)c(this)h(subsection)h(w)m(e)f (consider)g(a)f(concrete)h Fo(W)1932 3161 y Fw(\003)1971 3198 y Fu(-algebra)f Fp(M)g Fn(\032)g(B)s Fu(\()p Fn(H)q Fu(\))h(with)f(a)g Fo(W)3309 3161 y Fw(\003)3348 3198 y Fu(-dynamics)0 3318 y Fo(\034)48 b Fu(implemen)m(ted)38 b(b)m(y)f(a)f(self-adjoin)m(t)g(op)s(erator)g Fo(L)g Fu(and)g(assume)i(that)e Fo(Q)g Fu(is)h(a)e(selfadjoin)m(t)i(op)s (erator)0 3438 y(a\016liated)c(to)f Fp(M)p Fu(.)43 b(W)-8 b(e)33 b(form)m(ulate)g(the)g(follo)m(wing)g(assumption)h(on)f Fo(Q)p Fu(:)0 3642 y Fj(Assumption)38 b(3.A)49 b Fo(L)22 b Fu(+)g Fo(Q)36 b Fg(is)e(essential)5 b(ly)34 b(self-adjoint)g(on)h Fn(D)s Fu(\()p Fo(L)p Fu(\))22 b Fn(\\)g(D)s Fu(\()p Fo(Q)p Fu(\))p Fg(.)0 3870 y Fj(Theorem)75 b(3.3)49 b Fg(Supp)-5 b(ose)35 b(that)g(Assumption)f(3.A)h(holds)g(and)f(let)1315 4090 y Fo(\034)1368 4049 y Fl(t)1357 4115 y(Q)1417 4090 y Fu(\()p Fo(A)p Fu(\))28 b(=)f(e)1740 4049 y Fk(i)p Fl(t)p Fk(\()p Fl(L)p Fk(+)p Fl(Q)p Fk(\))2003 4090 y Fo(A)p Fu(e)2119 4049 y Fw(\000)p Fk(i)p Fl(t)p Fk(\()p Fl(L)p Fk(+)p Fl(Q)p Fk(\))2437 4090 y Fo(:)1066 b Fu(\(3.10\))0 4310 y Fg(Then)41 4462 y Fu(\(1\))48 b Fo(\034)256 4477 y Fl(Q)351 4462 y Fg(is)35 b(a)g Fo(W)647 4425 y Fw(\003)686 4462 y Fg(-dynamics)e(on)i Fp(M)p Fg(;)41 4623 y Fu(\(2\))48 b Fg(If)35 b Fo(Q)g Fg(is)g(b)-5 b(ounde)g(d,)34 b(then)g Fo(\034)1193 4638 y Fl(Q)1288 4623 y Fg(de\014ne)-5 b(d)34 b(by)h(\(3.10\))f(c)-5 b(oincides)33 b(with)i Fo(\034)2710 4638 y Fl(Q)2805 4623 y Fg(de\014ne)-5 b(d)34 b(by)h(\(3.6\).)0 4883 y Fj(Pro)s(of.)65 b Fu(Let)33 b Fo(A)28 b Fn(2)g Fp(M)p Fu(.)43 b(The)33 b(T)-8 b(rotter)33 b(pro)s(duct)g(form)m(ula)f (\(Theorem)i(A.1\))f(yields)h(that)805 5103 y Fo(\034)858 5062 y Fl(t)847 5128 y(Q)907 5103 y Fu(\()p Fo(A)p Fu(\))28 b(=)f(s)c Fn(\000)47 b Fu(lim)1348 5163 y Fl(n)p Fw(!1)1548 5022 y Fc(\000)1594 5103 y Fu(e)1637 5062 y Fk(i)p Fl(tL=n)1813 5103 y Fu(e)1856 5062 y Fk(i)p Fl(tQ=n)2039 5022 y Fc(\001)2084 5038 y Fl(n)2148 5103 y Fo(A)2238 5022 y Fc(\000)2283 5103 y Fu(e)2326 5062 y Fw(\000)p Fk(i)p Fl(tQ=n)2564 5103 y Fu(e)2607 5062 y Fw(\000)p Fk(i)p Fl(tL=n)2838 5022 y Fc(\001)2884 5038 y Fl(n)2947 5103 y Fo(:)1841 5753 y Fu(17)p eop %%Page: 18 18 18 17 bop 0 407 a Fu(Since)33 b(exp)q(\()p Fn(\006)p Fu(i)p Fo(tQ=n)p Fu(\))c Fn(2)f Fp(M)p Fu(,)k Fo(\034)1142 371 y Fl(t)1131 432 y(Q)1191 407 y Fu(\()p Fo(A)p Fu(\))27 b Fn(2)h Fp(M)p Fu(.)43 b(Therefore,)33 b Fo(\034)2146 422 y Fl(Q)2238 407 y Fu(is)g(a)e Fo(W)2522 371 y Fw(\003)2562 407 y Fu(-dynamics)i(and)f(\(1\))f(is)i(pro)m(v)m(en.)0 527 y(\(2\))f(follo)m(ws)h(from)g(Theorem)h(3.1)e(\(5\).)43 b Fe(2)146 697 y Fu(Under)34 b(Assumption)g(3.A)e(w)m(e)i(set)1428 917 y(E)1494 876 y Fl(\034)1494 942 y(Q)1554 917 y Fu(\()p Fo(t)p Fu(\))28 b(:=)f(e)1866 876 y Fk(i)p Fl(t)p Fk(\()p Fl(L)p Fk(+)p Fl(Q)p Fk(\))2129 917 y Fu(e)2172 876 y Fw(\000)p Fk(i)p Fl(tL)2325 917 y Fo(:)1178 b Fu(\(3.11\))0 1137 y(Again,)31 b(for)f(simplicit)m(y)j(w)m(e)e(will)h(often)e(write)h (E)1781 1152 y Fl(Q)1841 1137 y Fu(\()p Fo(t)p Fu(\))g(for)f(E)2196 1101 y Fl(\034)2196 1163 y(Q)2256 1137 y Fu(\()p Fo(t)p Fu(\).)43 b(By)31 b(the)g(T)-8 b(rotter)31 b(pro)s(duct)f(form)m(ula) 404 1374 y(E)470 1389 y Fl(Q)530 1374 y Fu(\()p Fo(t)p Fu(\))e(=)g(s)p Fn(\000)41 b Fu(lim)905 1434 y Fl(n)p Fw(!1)1106 1374 y Fu(exp)q(\(i)p Fo(tQ=n)p Fu(\))17 b(exp)q(\(i)p Fo(t\034)1896 1333 y Fl(t=n)2004 1374 y Fu(\()p Fo(Q)p Fu(\))p Fo(=n)p Fu(\))g Fn(\001)g(\001)g(\001)e Fu(exp)q(\(i)p Fo(t\034)2754 1333 y Fl(t)p Fk(\()p Fl(n)p Fw(\000)p Fk(1\))p Fl(=n)3050 1374 y Fu(\()p Fo(Q)p Fu(\))p Fo(=n)p Fu(\))p Fo(;)0 1627 y Fu(hence)34 b(E)337 1642 y Fl(Q)397 1627 y Fu(\()p Fo(t)p Fu(\))28 b Fn(2)g Fp(M)p Fu(.)0 1830 y Fj(Theorem)75 b(3.4)49 b Fg(Supp)-5 b(ose)31 b(that)h (Assumption)f(3.A)h(holds.)42 b(Then)31 b(al)5 b(l)31 b(the)g(statements)h(of)f(The)-5 b(or)g(em)0 1951 y(3.1)34 b(hold.)0 2239 y Fs(3.4)135 b(P)l(erturbations)46 b(of)g(Liouvilleans)0 2424 y Fu(W)-8 b(e)44 b(con)m(tin)m(ue)h(with)g(the)f(setup)h(of)e(the) h(previous)h(subsection.)79 b(In)44 b(addition,)j(w)m(e)e(supp)s(ose)g (that)0 2544 y(\()p Fp(M)p Fo(;)17 b Fn(H)q Fo(;)g(J)n(;)g Fn(H)497 2508 y Fk(+)555 2544 y Fu(\))33 b(is)g(a)f(standard)h(form)f (and)h(that)f Fo(L)h Fu(is)g(the)g(Liouvillean)h(of)e Fo(\034)11 b Fu(.)146 2665 y(De\014ne)1445 2785 y Fo(L)1511 2800 y Fl(Q)1599 2785 y Fu(:=)27 b Fo(L)c Fu(+)f Fo(Q)g Fn(\000)h Fo(J)9 b(QJ)n(:)1196 b Fu(\(3.12\))0 2960 y(W)-8 b(e)33 b(set)g(an)g(additional)f(h)m(yp)s(othesis:)0 3188 y Fj(Assumption)38 b(3.B)49 b Fg(The)29 b(op)-5 b(er)g(ator)29 b Fo(L)1475 3203 y Fl(Q)1564 3188 y Fg(is)h(essential)5 b(ly)28 b(self-adjoint)h(on)g Fn(D)s Fu(\()p Fo(L)p Fu(\))10 b Fn(\\)g(D)s Fu(\()p Fo(Q)p Fu(\))g Fn(\\)g(D)s Fu(\()p Fo(J)f(QJ)g Fu(\))p Fo(:)146 3416 y Fu(The)34 b(main)f(result)g(of)f (this)i(section)f(is:)0 3620 y Fj(Theorem)75 b(3.5)49 b Fg(Assume)34 b(that)g(Assumptions)g(3.A)g(and)f(3.B)g(hold.)44 b(Then)33 b Fo(L)2964 3635 y Fl(Q)3058 3620 y Fg(is)h(the)f(Liouvil)5 b(lian)0 3740 y(for)35 b Fo(\034)198 3755 y Fl(Q)258 3740 y Fg(.)0 3943 y Fj(Pro)s(of.)65 b Fu(W)-8 b(e)33 b(ha)m(v)m(e)h(to)e(sho)m(w)i(that)e(for)g Fo(t)c Fn(2)g Fm(R)p Fu(:)0 4064 y(\(1\))k Fo(\034)210 4028 y Fl(t)199 4089 y(Q)259 4064 y Fu(\()p Fo(A)p Fu(\))c(=)f(e)582 4028 y Fk(i)p Fl(tL)675 4039 y Fb(Q)732 4064 y Fo(A)p Fu(e)848 4028 y Fw(\000)p Fk(i)p Fl(tL)996 4039 y Fb(Q)1053 4064 y Fu(,)33 b Fo(A)28 b Fn(2)g Fp(M)p Fu(;)0 4209 y(\(2\))k(e)200 4173 y Fk(i)p Fl(tL)293 4184 y Fb(Q)350 4209 y Fn(H)435 4173 y Fk(+)522 4209 y Fn(\032)c(H)712 4173 y Fk(+)771 4209 y Fu(.)146 4329 y(Clearly)-8 b(,)1375 4450 y(e)1418 4409 y Fk(i)p Fl(tJ)6 b(QJ)1695 4450 y Fu(=)28 b Fo(J)9 b Fu(e)1905 4409 y Fw(\000)p Fk(i)p Fl(tQ)2065 4450 y Fo(J)37 b Fn(2)28 b Fp(M)2355 4409 y Fw(0)2378 4450 y Fo(:)1125 b Fu(\(3.13\))0 4624 y(By)27 b(de\014nition,)h Fn(D)s Fu(\()p Fo(L)9 b Fu(+)g Fo(Q)p Fu(\))27 b Fn(\033)h(D)s Fu(\()p Fo(L)p Fu(\))9 b Fn(\\)g(D)s Fu(\()p Fo(Q)p Fu(\).)41 b(Therefore,)29 b Fn(D)s Fu(\()p Fo(L)9 b Fu(+)g Fo(Q)p Fu(\))g Fn(\\)g(D)s Fu(\()p Fo(J)g(QJ)g Fu(\))27 b Fn(\033)h(D)s Fu(\()p Fo(L)p Fu(\))9 b Fn(\\)g(D)s Fu(\()p Fo(Q)p Fu(\))g Fn(\\)0 4745 y(D)s Fu(\()p Fo(J)g(QJ)g Fu(\).)42 b(Hence,)30 b(b)m(y)e(Hyp)s(othesis)h(3.B,)g Fo(L)1635 4760 y Fl(Q)1722 4745 y Fu(is)f(essen)m(tially)j(self-adjoin) m(t)c(on)h Fn(D)s Fu(\()p Fo(L)12 b Fu(+)g Fo(Q)p Fu(\))g Fn(\\)g(D)s Fu(\()p Fo(J)d(QJ)g Fu(\),)0 4865 y(and)33 b(w)m(e)g(can)g(use)h(the)f(T)-8 b(rotter)32 b(form)m(ula)h(\(Theorem)h (A.1\))e(to)h(write)1052 5086 y(e)1095 5050 y Fk(i)p Fl(tL)1188 5061 y Fb(Q)1328 5086 y Fu(=)28 b(s)22 b Fn(\000)47 b Fu(lim)1592 5146 y Fl(n)p Fw(!1)1793 5005 y Fc(\000)1838 5086 y Fu(e)1881 5050 y Fk(\(i)p Fl(tL)p Fk(+)p Fl(Q)p Fk(\))p Fl(=n)2222 5086 y Fo(e)2267 5050 y Fw(\000)p Fk(i)p Fl(tJ)6 b(QJ)r(=n)2591 5005 y Fc(\001)2636 5026 y Fl(n)2700 5086 y Fo(:)1841 5753 y Fu(18)p eop %%Page: 19 19 19 18 bop 0 407 a Fu(Therefore,)34 b(for)e(all)g Fo(A)c Fn(2)g Fp(M)p Fu(,)561 603 y Fo(\034)614 567 y Fl(t)603 628 y(Q)663 603 y Fu(\()p Fo(A)p Fu(\))83 b(=)28 b(e)1042 562 y Fk(i)p Fl(t)p Fk(\()p Fl(L)p Fk(+)p Fl(Q)p Fk(\))1305 603 y Fo(A)p Fu(e)1421 562 y Fw(\000)p Fk(i)p Fl(t)p Fk(\()p Fl(L)p Fk(+)p Fl(Q)p Fk(\))895 827 y Fu(=)g(s)p Fn(\000)42 b Fu(lim)1131 887 y Fl(n)p Fw(!1)1332 746 y Fc(\000)1378 827 y Fu(e)1421 786 y Fk(i)p Fl(t)p Fk(\()p Fl(L)p Fk(+)p Fl(Q)p Fk(\))p Fl(=n)1762 827 y Fo(e)1807 786 y Fw(\000)p Fk(i)p Fl(tJ)6 b(QJ)r(=n)2130 746 y Fc(\001)2176 762 y Fl(n)2239 827 y Fo(A)2329 746 y Fc(\000)2375 827 y Fu(e)2418 786 y Fk(i)p Fl(tJ)g(QJ)r(=n)2687 827 y Fu(e)2730 786 y Fw(\000)p Fk(i)p Fl(t)p Fk(\()p Fl(L)p Fk(+)p Fl(Q)p Fk(\))p Fl(=n)3126 746 y Fc(\001)3171 762 y Fl(n)895 1044 y Fu(=)28 b(e)1042 1003 y Fk(i)p Fl(tL)1135 1014 y Fb(Q)1192 1044 y Fo(A)p Fu(e)1308 1003 y Fw(\000)p Fk(i)p Fl(tL)1456 1014 y Fb(Q)1513 1044 y Fo(:)3530 821 y Fu(\(3.14\))0 1241 y(This)34 b(yields)g(\(1\).)146 1361 y(T)-8 b(o)33 b(establish)h(\(2\),)e(note)h(that)f(since)i(e)1586 1325 y Fk(i)p Fl(tQ)1724 1361 y Fu(and)f(e)1957 1325 y Fk(i)p Fl(tJ)6 b(QJ)2184 1361 y Fu(comm)m(ute)1403 1559 y(e)1446 1518 y Fk(i)p Fl(t)p Fk(\()p Fl(Q)p Fw(\000)p Fl(J)g(QJ)g Fk(\))1834 1559 y Fu(=)27 b(e)1980 1518 y Fk(i)p Fl(tQ)2086 1559 y Fo(J)9 b Fu(e)2192 1518 y Fk(i)p Fl(tQ)2297 1559 y Fo(J)n(:)0 1757 y Fu(Hence)1464 1878 y(e)1507 1837 y Fk(i)p Fl(t)p Fk(\()p Fl(Q)p Fw(\000)p Fl(J)d(QJ)g Fk(\))1867 1878 y Fn(H)1952 1837 y Fk(+)2039 1878 y Fn(\032)28 b(H)2229 1837 y Fk(+)2288 1878 y Fo(:)0 2043 y Fu(Moreo)m(v)m(er,)1595 2163 y(e)1638 2122 y Fk(i)p Fl(tL)1736 2163 y Fn(H)1821 2122 y Fk(+)1908 2163 y Fn(\032)g(H)2098 2122 y Fk(+)2157 2163 y Fo(:)0 2328 y Fu(By)47 b(de\014nition,)k Fn(D)s Fu(\()p Fo(Q)p Fu(\))32 b Fn(\\)g(D)s Fu(\()p Fo(J)9 b(QJ)g Fu(\))51 b Fn(\032)h(D)s Fu(\()p Fo(Q)32 b Fu(+)f Fo(J)9 b(QJ)g Fu(\).)86 b(Therefore,)52 b Fn(D)s Fu(\()p Fo(L)p Fu(\))31 b Fn(\\)h(D)s Fu(\()p Fo(Q)g Fn(\000)g Fo(J)9 b(QJ)g Fu(\))52 b Fn(\033)0 2448 y(D)s Fu(\()p Fo(L)p Fu(\))17 b Fn(\\)h(D)s Fu(\()p Fo(Q)p Fu(\))f Fn(\\)g(D)s Fu(\()p Fo(J)9 b(QJ)g Fu(\).)43 b(Hence)31 b Fo(L)1438 2463 y Fl(Q)1529 2448 y Fu(is)f(essen)m(tially)j (self-adjoin)m(t)d(on)g Fn(D)s Fu(\()p Fo(L)p Fu(\))17 b Fn(\\)h(D)s Fu(\()p Fo(Q)f Fn(\000)h Fo(J)9 b(QJ)g Fu(\))30 b(and)0 2569 y(it)j(follo)m(ws)g(from)f(Theorem)i(A.1)f(that) 1106 2767 y(e)1149 2726 y Fk(i)p Fl(tL)1242 2737 y Fb(Q)1327 2767 y Fu(=)28 b(s)22 b Fn(\000)47 b Fu(lim)1591 2827 y Fl(n)p Fw(!1)1791 2686 y Fc(\000)1837 2767 y Fu(e)1880 2726 y Fk(i)p Fl(tL=n)2056 2767 y Fu(e)2099 2726 y Fk(i)p Fl(t)p Fk(\()p Fl(Q)p Fw(\000)p Fl(J)6 b(QJ)g Fk(\))p Fl(=n)2537 2686 y Fc(\001)2582 2702 y Fl(n)2646 2767 y Fo(:)0 2999 y Fu(This)34 b(and)e(the)h(fact)g(that)f Fn(H)1069 2963 y Fk(+)1161 2999 y Fu(is)h(a)f(closed)i(set)f(imply)h (\(2\).)43 b Fe(2)146 3163 y Fu(The)34 b(follo)m(wing)f(form)m(ulas)g (are)f(sometimes)j(useful:)0 3347 y Fj(Theorem)75 b(3.6)90 b Fu(\(1\))49 b Fg(Assume)35 b(that)g(Assumptions)g(3.A)f(and)h(3.B)f (hold.)44 b(Then)34 b(for)h Fo(t)28 b Fn(2)g Fm(R)p Fg(,)1343 3549 y Fu(E)1409 3564 y Fl(Q)1469 3549 y Fu(\()p Fo(t)p Fu(\))g(=)f(e)1754 3508 y Fk(i)p Fl(tL)1847 3519 y Fb(Q)1904 3549 y Fu(e)1947 3508 y Fw(\000)p Fk(i)p Fl(t)p Fk(\()p Fl(L)p Fw(\000)p Fl(J)6 b(QJ)g Fk(\))2354 3549 y Fo(;)1387 3776 y Fu(e)1430 3735 y Fk(i)p Fl(tL)1523 3746 y Fb(Q)1608 3776 y Fu(=)27 b Fo(J)9 b Fu(E)1840 3791 y Fl(Q)1900 3776 y Fu(\()p Fo(t)p Fu(\))p Fo(J)g Fu(e)2117 3735 y Fk(i)p Fl(tL)2215 3776 y Fu(E)2281 3791 y Fl(Q)2341 3776 y Fu(\()p Fn(\000)p Fo(t)p Fu(\))2529 3735 y Fw(\000)p Fk(1)2624 3776 y Fo(:)41 4011 y Fu(\(2\))48 b Fg(Assume)35 b(that)h Fo(Q)f Fg(is)f Fo(\034)11 b Fg(-analytic.)45 b(Then)34 b(for)h Fo(z)d Fn(2)c Fm(C)p Fg(,)1316 4213 y Fu(E)1382 4228 y Fl(Q)1442 4213 y Fu(\()p Fo(z)t Fu(\))g(=)g(e)1742 4172 y Fk(i)p Fl(z)s(L)1846 4183 y Fb(Q)1902 4213 y Fu(e)1945 4172 y Fw(\000)p Fk(i)p Fl(z)s Fk(\()p Fl(L)p Fw(\000)p Fl(J)6 b(QJ)g Fk(\))2362 4213 y Fo(;)1365 4439 y Fu(e)1408 4398 y Fk(i)p Fl(z)s(L)1512 4409 y Fb(Q)1595 4439 y Fu(=)28 b Fo(J)9 b Fu(E)1828 4454 y Fl(Q)1888 4439 y Fu(\()p 1926 4385 50 4 v Fo(z)c Fu(\))p Fo(J)k Fu(e)2120 4398 y Fk(i)p Fl(z)s(L)2227 4439 y Fu(E)2293 4454 y Fl(Q)2354 4439 y Fu(\()p Fn(\000)p Fo(z)t Fu(\))2556 4398 y Fw(\000)p Fk(1)2651 4439 y Fo(:)0 4808 y Ft(4)161 b(Relativ)l(e)53 b(mo)t(dular)h(theory)f(and)g(relativ)l(e)g(en)l(trop)l(y)0 5027 y Fu(One)23 b(of)g(the)g(main)g(to)s(ols)g(used)h(in)f(our)g(pap)s (er)g(is)h(the)f(relativ)m(e)h(mo)s(dular)f(theory)h(and)f(relativ)m(e) h(en)m(trop)m(y)-8 b(.)0 5147 y(W)g(e)30 b(dev)m(ote)h(this)g(section)g (to)e(a)h(concise)h(in)m(tro)s(duction)g(to)e(this)h(sub)5 b(ject.)45 b(Our)29 b(presen)m(tation)j(follo)m(ws)0 5268 y(partly)h([Ar4,)f(Ar5,)h(Don)o(,)g(Uh,)g(OP].)1841 5753 y(19)p eop %%Page: 20 20 20 19 bop 0 407 a Fs(4.1)135 b(Relativ)l(e)48 b(mo)t(dular)d(op)t (erator)0 592 y Fu(Let)39 b Fp(M)f Fn(\032)i(B)s Fu(\()p Fn(H)q Fu(\))f(b)s(e)g(a)g Fo(W)1042 555 y Fw(\003)1081 592 y Fu(-algebra.)62 b(Let)40 b(\010)p Fo(;)17 b Fu(\011)38 b Fn(2)h(H)q Fu(.)63 b(F)-8 b(ollo)m(wing)40 b(Araki)f([Ar5],)i(w)m(e)f (de\014ne)g(the)0 712 y(op)s(erator)32 b Fo(S)453 727 y Fk(\010)p Fl(;)p Fk(\011)615 712 y Fu(on)h(domain)f Fp(M)p Fu(\011)22 b(+)g(\(1)g Fn(\000)g Fu(s)1644 676 y Fw(0)1644 737 y Fk(\011)1704 712 y Fu(\))p Fn(H)33 b Fu(b)m(y)1365 880 y Fo(S)1425 895 y Fk(\010)p Fl(;)p Fk(\011)1555 880 y Fu(\()p Fo(A)p Fu(\011)22 b(+)g(\002\))27 b(=)h(s)2145 895 y Fk(\011)2204 880 y Fo(A)2277 838 y Fw(\003)2317 880 y Fu(\010)p Fo(;)0 1047 y Fu(where)h Fo(A)f Fn(2)g Fp(M)f Fu(and)h(\002)g Fn(2)g Fu(\(1)13 b Fn(\000)g Fu(s)1215 1011 y Fw(0)1215 1072 y Fk(\011)1274 1047 y Fu(\))p Fn(H)28 b Fu(=)g(\()p Fp(M)p Fu(\011\))1785 1011 y Fw(?)1843 1047 y Fu(.)42 b(It)28 b(is)g(easy)h(to)f(c)m(hec)m(k) i(that)e Fo(S)2956 1062 y Fk(\010)p Fl(;)p Fk(\011)3113 1047 y Fu(is)g(a)g(w)m(ell)h(de\014ned)0 1168 y(an)m(tilinear)k (closable)h(op)s(erator.)43 b(Its)33 b(closure)h(will)f(b)s(e)g (denoted)g(b)m(y)h(the)f(same)g(sym)m(b)s(ol.)146 1288 y(It)g(is)g(useful)h(to)e(note)h(that)1180 1456 y Fp(M)p Fu(\011)27 b(=)g Fn(f)p Fo(A)p Fu(\011)h(:)f Fo(A)h Fn(2)g Fp(M)p Fo(;)49 b(A)p Fu(s)2259 1471 y Fk(\011)2346 1456 y Fu(=)28 b Fo(A)p Fn(g)p Fo(;)0 1623 y Fu(and)33 b(that)f(for)g Fo(A)c Fn(2)g Fp(M)k Fu(satisfying)i Fo(A)p Fu(s)1428 1638 y Fk(\011)1515 1623 y Fu(=)28 b Fo(A)k Fu(and)h(\002)f(as)h(ab)s (o)m(v)m(e)g(w)m(e)h(ha)m(v)m(e)1414 1791 y Fo(S)1474 1806 y Fk(\010)p Fl(;)p Fk(\011)1603 1791 y Fu(\()p Fo(A)p Fu(\011)23 b(+)f(\002\))27 b(=)h Fo(A)2229 1750 y Fw(\003)2268 1791 y Fu(\010)p Fo(:)1165 b Fu(\(4.15\))146 1959 y(The)34 b(p)s(ositiv)m(e)g(op)s(erator)1529 2079 y(\001)1610 2094 y Fk(\010)p Fl(;)p Fk(\011)1768 2079 y Fu(=)27 b Fo(S)1937 2038 y Fw(\003)1931 2104 y Fk(\010)p Fl(;)p Fk(\011)2061 2079 y Fo(S)2121 2094 y Fk(\010)p Fl(;)p Fk(\011)0 2231 y Fu(will)33 b(b)s(e)g(called)g(the)g(relativ)m(e)h(mo)s (dular)f(op)s(erator.)43 b(The)33 b(follo)m(wing)g(facts)g(are)f(pro)m (v)m(en)i(in)f([Ar5]:)0 2400 y Fj(Theorem)75 b(4.1)90 b Fu(\(1\))49 b(Ker)17 b(\001)1174 2415 y Fk(\010)p Fl(;)p Fk(\011)1331 2400 y Fu(=)28 b(Ker)16 b(s)1646 2364 y Fw(0)1646 2425 y Fk(\011)1706 2400 y Fu(s)1744 2415 y Fk(\010)1800 2400 y Fu(;)41 2565 y(\(2\))48 b(\001)295 2580 y Fl(\025)p Fk(\010)p Fl(;\026)p Fk(\011)537 2565 y Fu(=)651 2525 y Fl(\025)692 2502 y Ff(2)p 650 2542 78 4 v 650 2599 a Fl(\026)692 2580 y Ff(2)737 2565 y Fu(\001)818 2580 y Fk(\010)p Fl(;)p Fk(\011)948 2565 y Fg(,)35 b Fo(\025;)17 b(\026)27 b Fn(2)h Fm(R)p Fg(;)41 2729 y Fu(\(3\))48 b Fg(if)35 b Fo(B)40 b Fg(b)-5 b(elongs)34 b(to)h(the)g(c)-5 b(enter)34 b(of)h Fp(M)p Fg(,)f(then)h Fo(B)40 b Fg(c)-5 b(ommutes)34 b(with)h Fu(\001)2689 2744 y Fk(\010)p Fl(;)p Fk(\011)2819 2729 y Fg(.)0 2970 y(In)40 b(the)h(r)-5 b(emaining)40 b(p)-5 b(art)41 b(of)f(the)h(the)-5 b(or)g(em)41 b(we)f(assume)g(that)h Fu(\()p Fp(M)p Fo(;)17 b Fn(H)q Fo(;)g(J)n(;)g Fn(H)2837 2934 y Fk(+)2896 2970 y Fu(\))40 b Fg(is)h(a)g(standar)-5 b(d)40 b(form)0 3090 y(and)34 b Fu(\010)p Fo(;)17 b Fu(\011)28 b Fn(2)g(H)586 3054 y Fk(+)645 3090 y Fg(.)45 b(Then)41 3242 y Fu(\(4\))j Fo(S)274 3257 y Fk(\010)p Fl(;)p Fk(\011)432 3242 y Fu(=)27 b Fo(J)9 b Fu(\001)679 3191 y Fk(1)p Fl(=)p Fk(2)679 3269 y(\010)p Fl(;)p Fk(\011)809 3242 y Fu(;)41 3425 y(\(5\))48 b(\001)295 3374 y Fk(1)p Fl(=)p Fk(2)295 3452 y(\010)p Fl(;)p Fk(\011)425 3425 y Fu(\011)28 b(=)f(\001)713 3374 y Fk(1)p Fl(=)p Fk(2)713 3452 y(\010)p Fl(;)p Fk(\011)843 3425 y Fu(s)881 3440 y Fk(\010)937 3425 y Fu(\011)h(=)f(s)1182 3389 y Fw(0)1182 3450 y Fk(\011)1242 3425 y Fu(\010;)41 3579 y(\(6\))48 b Fo(J)9 b Fu(\001)358 3594 y Fk(\011)p Fl(;)p Fk(\010)489 3579 y Fo(J)g Fu(\001)633 3594 y Fk(\010)p Fl(;)p Fk(\011)790 3579 y Fu(=)28 b(\001)975 3594 y Fk(\010)p Fl(;)p Fk(\011)1105 3579 y Fo(J)9 b Fu(\001)1249 3594 y Fk(\011)p Fl(;)p Fk(\010)1379 3579 y Fo(J)37 b Fu(=)27 b(s)1611 3543 y Fw(0)1611 3604 y Fk(\011)1671 3579 y Fu(s)1709 3594 y Fk(\010)1764 3579 y Fo(:)146 3779 y Fu(The)34 b(follo)m(wing)f(con)m(v)m(ergence)i(prop)s(ert)m(y)e(of)f (relativ)m(e)i(mo)s(dular)f(op)s(erators)f(will)h(b)s(e)g(useful.)0 3947 y Fj(Theorem)75 b(4.2)49 b Fg(L)-5 b(et)39 b Fu(\()p Fp(M)p Fo(;)17 b Fn(H)q Fo(;)g(J)n(;)g Fn(H)1373 3911 y Fk(+)1431 3947 y Fu(\))39 b Fg(b)-5 b(e)38 b(a)g(standar)-5 b(d)38 b(form.)55 b(Supp)-5 b(ose)37 b(that)i Fu(\011)3059 3962 y Fl(n)3106 3947 y Fo(;)17 b Fu(\010)3220 3962 y Fl(n)3301 3947 y Fn(2)35 b(H)3487 3911 y Fk(+)3546 3947 y Fg(,)k(that)0 4068 y Fu(\001)81 4083 y Fk(\010)132 4091 y Fb(n)175 4083 y Fl(;)p Fk(\011)250 4091 y Fb(n)330 4068 y Fn(!)33 b Fo(M)50 b Fg(in)37 b(the)i(str)-5 b(ong)38 b(r)-5 b(esolvent)37 b(sense,)i(and)e(that)i Fu(w)q Fn(\000)18 b Fu(lim)2591 4083 y Fl(n)2655 4068 y Fu(\011)2731 4083 y Fl(n)2812 4068 y Fu(=)34 b(\011)p Fg(,)39 b Fu(s)p Fn(\000)17 b Fu(lim)3335 4083 y Fl(n)3398 4068 y Fu(s)3436 4083 y Fk(\011)3491 4091 y Fb(n)3572 4068 y Fu(=)34 b(s)3720 4083 y Fk(\011)0 4188 y Fg(and)g Fu(w)q Fn(\000)18 b Fu(lim)491 4203 y Fl(n)554 4188 y Fu(\010)624 4203 y Fl(n)699 4188 y Fu(=)28 b(\010)p Fg(.)45 b(Then)34 b Fo(M)39 b Fu(=)27 b(\001)1519 4203 y Fk(\010)p Fl(;)p Fk(\011)1649 4188 y Fg(.)0 4357 y Fj(Pro)s(of.)65 b Fu(F)-8 b(or)32 b Fo(A)c Fn(2)g Fp(M)p Fu(,)1348 4477 y(\001)1429 4426 y Fk(1)p Fl(=)p Fk(2)1429 4504 y(\010)1480 4512 y Fb(n)1523 4504 y Fl(;)p Fk(\011)1598 4512 y Fb(n)1644 4477 y Fo(A)p Fu(\011)1793 4492 y Fl(n)1868 4477 y Fu(=)f Fo(J)9 b Fu(s)2072 4492 y Fk(\011)2127 4500 y Fb(n)2174 4477 y Fo(A)2247 4436 y Fw(\003)2287 4477 y Fu(\010)2357 4492 y Fl(n)2404 4477 y Fo(:)0 4629 y Fu(Note)34 b(that)g Fo(A)p Fu(\011)599 4644 y Fl(n)676 4629 y Fn(!)c Fo(A)p Fu(\011)k(w)m(eakly)i(and)e Fo(J)9 b Fu(s)1605 4644 y Fk(\011)1660 4652 y Fb(n)1707 4629 y Fo(A)1780 4593 y Fw(\003)1820 4629 y Fu(\010)1890 4644 y Fl(n)1967 4629 y Fn(!)30 b Fo(J)9 b Fu(s)2198 4644 y Fk(\011)2258 4629 y Fo(A)2331 4593 y Fw(\003)2370 4629 y Fu(\010)35 b(w)m(eakly)-8 b(.)49 b(Hence,)36 b(b)m(y)f(Prop)s(osition)0 4749 y(A.6)e(and)f (remark)h(after)g(it,)f Fo(A)p Fu(\011)c Fn(2)g(D)s Fu(\()p Fo(M)10 b Fu(\))33 b(and)1512 4917 y Fo(M)10 b(A)p Fu(\011)28 b(=)g Fo(J)9 b Fu(s)1998 4932 y Fk(\011)2058 4917 y Fo(A)2131 4876 y Fw(\003)2170 4917 y Fu(\010)p Fo(:)146 5085 y Fu(No)m(w)40 b(let)g(\002)f Fn(2)g Fu(\(1)27 b Fn(\000)g Fu(s)999 5049 y Fw(0)999 5110 y Fk(\011)1058 5085 y Fu(\))p Fn(H)40 b Fu(and)g(\002)1493 5100 y Fl(n)1579 5085 y Fu(:=)f(\(1)26 b Fn(\000)h Fu(s)1976 5049 y Fw(0)1976 5110 y Fk(\011)2031 5118 y Fb(n)2078 5085 y Fu(\)\002.)64 b(Since)40 b(s)2582 5049 y Fw(0)2582 5110 y Fk(\011)2637 5118 y Fb(n)2723 5085 y Fn(!)f Fu(s)2900 5049 y Fw(0)2900 5110 y Fk(\011)2999 5085 y Fu(strongly)-8 b(,)41 b(\002)3479 5100 y Fl(n)3565 5085 y Fn(!)e Fu(\002)0 5205 y(strongly)-8 b(.)44 b(Since)34 b(\001)743 5220 y Fk(\010)794 5228 y Fb(n)836 5220 y Fl(;)p Fk(\011)911 5228 y Fb(n)958 5205 y Fu(\002)1034 5220 y Fl(n)1108 5205 y Fu(=)28 b(0,)k(\002)c Fn(2)g(D)s Fu(\()p Fo(M)10 b Fu(\))32 b(and)h Fo(M)10 b Fu(\002)28 b(=)g(0.)43 b(This)34 b(yields)g Fo(M)39 b Fu(=)27 b(\001)3245 5220 y Fk(\010)p Fl(;)p Fk(\011)3375 5205 y Fu(.)44 b Fe(2)1841 5753 y Fu(20)p eop %%Page: 21 21 21 20 bop 0 407 a Fs(4.2)135 b(Relativ)l(e)48 b(en)l(trop)l(y)0 592 y Fu(Let)41 b Fp(M)g Fu(b)s(e)h(a)f Fo(W)667 555 y Fw(\003)706 592 y Fu(-algebra.)69 b(The)42 b(relativ)m(e)g(en)m(trop) m(y)h(of)e(t)m(w)m(o)h(functionals)g Fo( )t(;)17 b(\036)42 b Fn(2)g Fp(M)3314 555 y Fk(+)3314 616 y Fw(\003)3373 592 y Fu(,)i(denoted)0 712 y(En)m(t)q(\()p Fo( )t Fn(j)p Fo(\036)p Fu(\),)30 b(is)h(de\014ned)i(as)d(follo)m(ws.)44 b(Cho)s(ose)32 b(a)e(standard)h(form)g(\()p Fo(\031)t(;)17 b Fn(H)q Fo(;)g(J)n(;)g Fn(H)2850 676 y Fk(+)2908 712 y Fu(\))31 b(of)f Fp(M)g Fu(and)h(let)g(\011,)g(\010,)0 832 y(b)s(e)i(the)g(standard)g(v)m(ector)g(represen)m(tativ)m(es)j(of)c Fo( )t Fu(,)h Fo(\036)p Fu(.)43 b(Then)1005 1164 y(En)m(t)q(\()p Fo( )t Fn(j)p Fo(\036)p Fu(\))27 b(=)1520 960 y Fc(8)1520 1050 y(<)1520 1229 y(:)1609 1061 y Fu(\(\011)p Fn(j)17 b Fu(log)g(\001)1991 1076 y Fk(\010)p Fl(;)p Fk(\011)2121 1061 y Fu(\011\))97 b(if)32 b(s)2459 1076 y Fl( )2540 1061 y Fn(\024)c Fo(s)2691 1076 y Fl(\036)2737 1061 y Fo(;)1609 1276 y Fn(\0001)546 b Fu(otherwise)q Fo(:)0 1502 y Fu(The)36 b(relativ)m(e)h(en)m(trop)m(y)g(w)m(as)g(in)m(tro)s (duced)f(b)m(y)h(Araki)f(in)g(fundamen)m(tal)g(pap)s(ers)g([Ar4,)g (Ar5].)52 b(In)36 b(the)0 1622 y(ab)s(o)m(v)m(e)c(de\014nition)f(w)m(e) h(used)g(the)f(sign)g(and)g(ordering)g(con)m(v)m(en)m(tion)i(of)d ([BR2].)42 b(The)32 b(relativ)m(e)g(en)m(trop)m(y)0 1743 y(is)h(discussed)i(in)e(detail)g(in)g(the)g(monograph)f([OP)q(].)146 1863 y(W)-8 b(e)33 b(will)h(need)f(the)g(follo)m(wing)g(w)m(ell-kno)m (wn)i(facts)e([Ar4,)f(Ar5,)h(OP,)f(Don].)0 2091 y Fj(Theorem)75 b(4.3)90 b Fu(\(1\))1128 2239 y(En)m(t\()p Fo( )t Fn(j)p Fo(\036)p Fu(\))27 b(=)h(lim)1663 2302 y Fl(t)p Fw(#)p Fk(0)1795 2239 y Fo(t)1830 2198 y Fw(\000)p Fk(1)1941 2129 y Fc(\020)2001 2239 y Fn(k)p Fu(\001)2132 2188 y Fl(t=)p Fk(2)2132 2266 y(\010)p Fl(;)p Fk(\011)2261 2239 y Fu(\011)p Fn(k)2387 2198 y Fk(2)2449 2239 y Fn(\000)22 b(k)p Fu(\011)p Fn(k)2724 2198 y Fk(2)2763 2129 y Fc(\021)2839 2239 y Fu(;)664 b(\(4.16\))41 2505 y(\(2\))48 b Fg(for)35 b Fo(\026;)17 b(\025)27 b Fn(2)h Fm(R)723 2469 y Fk(+)782 2505 y Fg(,)964 2725 y Fu(En)m(t)q(\()p Fo(\025 )t Fn(j)p Fo(\026\036)p Fu(\))e(=)i Fo(\025)p Fu(En)m(t)q(\()p Fo( )t Fn(j)p Fo(\036)p Fu(\))21 b(+)h Fo(\025 )t Fu(\(1\)\(log)17 b Fo(\026)22 b Fn(\000)h Fu(log)17 b Fo(\025)p Fu(\);)41 2986 y(\(3\))1180 3107 y(En)m(t\()p Fo( )t Fn(j)p Fo(\036)p Fu(\))27 b Fn(\024)h Fo( )t Fu(\(1\)\(log)18 b Fo(\036)p Fu(\(s)2203 3122 y Fl( )2255 3107 y Fu(\))23 b Fn(\000)f Fu(log)c Fo( )t Fu(\(1\)\))p Fo(;)214 3281 y Fg(in)35 b(p)-5 b(articular,)35 b(if)f Fo(\036)p Fu(\(s)1038 3296 y Fl( )1091 3281 y Fu(\))28 b(=)f Fo( )t Fu(\(1\))35 b Fg(then)1701 3501 y Fu(En)m(t\()p Fo( )t Fn(j)p Fo(\036)p Fu(\))27 b Fn(\024)i Fu(0;)41 3763 y(\(4\))48 b Fg(if)35 b Fo(Q)g Fg(is)g(a)g(self-adjoint)e(element)h(in)h(the)g(c)-5 b(enter)34 b(of)h Fp(M)f Fg(and)h Fo( )t Fu(\(1\))27 b(=)h(1)p Fg(,)34 b(then)1365 3983 y Fu(En)m(t)q(\()p Fo( )t Fn(j)p Fo(\036)p Fu(\))21 b(+)h Fo( )t Fu(\()p Fo(Q)p Fu(\))28 b Fn(\024)g Fu(log)18 b Fo(\036)p Fu(\(e)2504 3942 y Fl(Q)2564 3983 y Fu(\))p Fo(:)0 4311 y Fj(Pro)s(of.)82 b Fu(\(1\))41 b(Assume)i(\014rst)f(that)e(s)1388 4326 y Fk(\011)1490 4311 y Fn(\024)j Fu(s)1648 4326 y Fk(\010)1704 4311 y Fu(.)69 b(Then)42 b(the)g(statemen)m(t)g(follo)m(ws)g(from)f (the)h(sp)s(ectral)0 4431 y(theorem,)33 b(monotone)g(con)m(v)m(ergence) j(theorem)d(and)g(the)g(fact)f(that)1405 4651 y(lim)1425 4714 y Fl(t)p Fw(#)p Fk(0)1557 4651 y Fo(t)1592 4610 y Fw(\000)p Fk(1)1687 4651 y Fu(\()p Fo(x)1780 4610 y Fl(t)1832 4651 y Fn(\000)22 b Fu(1\))28 b(=)f(log)18 b Fo(x;)0 4914 y Fu(monotonically)35 b(on)f(the)g(in)m(terv)-5 b(als)35 b(0)30 b Fn(\024)h Fo(x)g Fn(\024)f Fu(1,)k Fo(x)d Fn(\025)g Fu(1.)47 b(If)34 b(s)2275 4929 y Fk(\010)2331 4914 y Fu(\011)c Fn(6)p Fu(=)g(\011,)k(then)h(\011)30 b(=)g(\011)3192 4929 y Fk(1)3254 4914 y Fu(+)23 b(\011)3429 4929 y Fk(2)3469 4914 y Fu(,)34 b(where)0 5034 y(\011)76 5049 y Fk(1)143 5034 y Fn(6)p Fu(=)28 b(0,)h(\011)428 5049 y Fk(1)495 5034 y Fn(?)f Fu(\011)676 5049 y Fk(2)744 5034 y Fu(and)h(\011)1006 5049 y Fk(1)1073 5034 y Fn(2)f Fu(Ker\001)1405 5049 y Fk(\010)p Fl(;)p Fk(\011)1535 5034 y Fu(,)h(and)g(one)g(easily)h(sho)m(ws)g(that)e(the)h(limit)g(in)g (\(4.16\))f(is)h Fn(\0001)p Fu(.)146 5155 y(Scaling)k(prop)s(ert)m(y)h (of)e(Theorem)i(4.1)e(yields)i(\(2\).)1841 5753 y(21)p eop %%Page: 22 22 22 21 bop 146 407 a Fu(W)-8 b(e)33 b(\014rst)g(pro)m(v)m(e)h(the)f (part)g(\(3\))f(under)h(the)g(assumption)h Fo(\036)p Fu(\(s)2409 422 y Fl( )2462 407 y Fu(\))27 b(=)h Fo( )t Fu(\(1\))f(=)h(1.)43 b(Using)1409 591 y(log)18 b Fo(x)28 b Fn(\024)g Fo(x)23 b Fn(\000)g Fu(1)p Fo(;)114 b(x)28 b(>)f Fu(0)p Fo(;)1160 b Fu(\(4.17\))0 774 y(w)m(e)34 b(get)1442 895 y(log)18 b(\001)1666 910 y Fk(\010)p Fl(;)p Fk(\011)1824 895 y Fn(\024)28 b Fu(\001)2010 910 y Fk(\010)p Fl(;)p Fk(\011)2162 895 y Fn(\000)22 b Fu(1)p Fo(:)0 1054 y Fu(Th)m(us)812 1174 y(En)m(t)q(\()p Fo( )t Fn(j)p Fo(\036)p Fu(\))27 b Fn(\024)h(k)p Fu(\001)1460 1123 y Fk(1)p Fl(=)p Fk(2)1460 1201 y(\010)p Fl(;)p Fk(\011)1589 1174 y Fu(\011)p Fn(k)1715 1133 y Fk(2)1777 1174 y Fn(\000)22 b(k)p Fu(\011)p Fn(k)2052 1133 y Fk(2)2119 1174 y Fu(=)27 b Fo(\036)p Fu(\(s)2356 1189 y Fl( )2409 1174 y Fu(\))22 b Fn(\000)h Fo( )t Fu(\(1\))k(=)h(0)p Fo(:)0 1361 y Fu(\(W)-8 b(e)33 b(used)h(\001)510 1310 y Fk(1)p Fl(=)p Fk(2)510 1388 y(\010)p Fl(;)p Fk(\011)640 1361 y Fu(\011)27 b(=)h(s)885 1325 y Fw(0)885 1386 y Fk(\011)944 1361 y Fu(\010)g(=)g Fo(J)9 b Fu(s)1247 1376 y Fk(\011)1306 1361 y Fu(\010\).)146 1482 y(T)-8 b(o)33 b(extend)h(\(3\))e(to)g(arbitrary)h Fo(\036)p Fu(,)f Fo( )t Fu(,)h(use)g(\(2\).)146 1602 y(T)-8 b(o)33 b(pro)m(v)m(e)h(\(4\),)e(note)h(that)f(since)i(e)1447 1566 y Fl(Q)1540 1602 y Fu(comm)m(utes)g(with)f(\001)2301 1617 y Fk(\010)p Fl(;)p Fk(\011)777 1816 y Fu(log)17 b(\001)1000 1831 y Fk(\010)p Fl(;)p Fk(\011)1152 1816 y Fu(+)22 b Fo(Q)h Fn(\000)f Fu(log)c Fo(\036)p Fu(\(e)1731 1775 y Fl(Q)1791 1816 y Fu(s)1829 1831 y Fl( )1882 1816 y Fu(\))27 b(=)h(log)2193 1706 y Fc(\020)2253 1816 y Fu(\001)2334 1831 y Fk(\010)p Fl(;)p Fk(\011)2464 1816 y Fu(e)2507 1775 y Fl(Q)2567 1816 y Fo(=\036)p Fu(\(e)2755 1775 y Fl(Q)2814 1816 y Fu(s)2852 1831 y Fl( )2905 1816 y Fu(\))2943 1706 y Fc(\021)0 2027 y Fu(The)34 b(inequalit)m(y)g (\(4.17\))e(yields)903 2238 y(log)1046 2128 y Fc(\020)1105 2238 y Fu(\001)1186 2253 y Fk(\010)p Fl(;)p Fk(\011)1316 2238 y Fu(e)1359 2197 y Fl(Q)1419 2238 y Fo(=\036)p Fu(\(e)1607 2197 y Fl(Q)1667 2238 y Fu(s)1705 2253 y Fl( )1758 2238 y Fu(\))1796 2128 y Fc(\021)1883 2238 y Fn(\024)c Fu(\001)2069 2253 y Fk(\010)p Fl(;)p Fk(\011)2199 2238 y Fu(e)2242 2197 y Fl(Q)2302 2238 y Fo(=\036)p Fu(\(e)2490 2197 y Fl(Q)2550 2238 y Fu(s)2588 2253 y Fl( )2641 2238 y Fu(\))22 b Fn(\000)g Fu(1)p Fo(:)0 2443 y Fu(Hence)552 2615 y(En)m(t)q(\()p Fo( )t Fn(j)p Fo(\036)p Fu(\))f(+)h Fo( )t Fu(\()p Fo(Q)p Fu(\))g Fn(\000)h Fu(log)17 b Fo(\036)p Fu(\(e)1679 2574 y Fl(Q)1739 2615 y Fu(s)1777 2630 y Fl( )1830 2615 y Fu(\))28 b Fn(\024)g(k)p Fu(\001)2132 2564 y Fk(1)p Fl(=)p Fk(2)2132 2642 y(\010)p Fl(;)p Fk(\011)2262 2615 y Fu(e)2305 2574 y Fl(Q=)p Fk(2)2435 2615 y Fu(\011)p Fn(k)2561 2574 y Fk(2)2600 2615 y Fo(=\036)p Fu(\(e)2788 2574 y Fl(Q)2848 2615 y Fu(s)2886 2630 y Fl( )2939 2615 y Fu(\))22 b Fn(\000)h Fu(1)1896 2850 y(=)k Fn(k)p Fu(e)2092 2809 y Fl(Q=)p Fk(2)2223 2850 y Fu(s)2261 2809 y Fw(0)2261 2875 y Fl( )2314 2850 y Fu(\010)p Fn(k)2434 2809 y Fk(2)2474 2850 y Fo(=\036)p Fu(\(e)2662 2809 y Fl(Q)2721 2850 y Fu(s)2759 2865 y Fl( )2812 2850 y Fu(\))22 b Fn(\000)h Fu(1)k(=)h(0)p Fo(;)0 3045 y Fu(where)34 b(w)m(e)f(used)h Fn(k)p Fu(e)741 3009 y Fl(Q=)p Fk(2)872 3045 y Fu(s)910 3009 y Fw(0)910 3071 y Fl( )963 3045 y Fu(\010)p Fn(k)28 b Fu(=)f Fn(k)p Fu(e)1307 3009 y Fl(Q=)p Fk(2)1438 3045 y Fo(J)9 b Fu(s)1539 3060 y Fl( )1592 3045 y Fo(J)g Fu(\010)p Fn(k)28 b Fu(=)f Fn(k)p Fo(J)9 b Fu(e)2062 3009 y Fl(Q=)p Fk(2)2193 3045 y Fo(s)2239 3060 y Fl( )2292 3045 y Fu(\010)p Fn(k)28 b Fu(=)f Fn(k)p Fu(e)2636 3009 y Fl(Q=)p Fk(2)2767 3045 y Fo(s)2813 3060 y Fl( )2865 3045 y Fu(\010)p Fn(k)p Fu(.)44 b Fe(2)0 3368 y Fs(4.3)135 b(Uhlmann's)46 b(monotonicit)l(y)g (theorem)0 3553 y Fu(In)29 b(this)h(subsection)g(w)m(e)g(pro)m(v)m(e)g (a)f(relativ)m(e)h(en)m(trop)m(y)g(inequalit)m(y)g(due)g(to)e(Uhlmann)i ([Uh].)43 b(Our)28 b(pro)s(of)0 3674 y(follo)m(ws)41 b(the)g(steps)g(of)f(an)g(argumen)m(t)h(in)f([OP)q(])g(and)g(is)h (based)g(on)f(an)g(in)m(terp)s(olation)h(theorem)g(for)0 3794 y(self-adjoin)m(t)34 b(op)s(erators)g(\(Theorem)i(A.7)e(in)g(the)h (app)s(endix\).)49 b(A)34 b(di\013eren)m(t)h(pro)s(of)e(can)i(b)s(e)f (found)g(in)0 3914 y([PuW)-8 b(o)q(].)146 4035 y(Let)37 b Fp(M)430 4050 y Fk(1)505 4035 y Fu(and)f Fp(M)803 4050 y Fk(2)878 4035 y Fu(b)s(e)g Fo(W)1120 3999 y Fw(\003)1159 4035 y Fu(-algebras.)54 b(A)36 b(map)h Fo(\015)h Fu(:)c Fp(M)2205 4050 y Fk(1)2278 4035 y Fn(7!)f Fp(M)2516 4050 y Fk(2)2591 4035 y Fu(is)j(called)h(a)f(Sc)m(h)m(w)m(artz)i(map)f (i\013)0 4155 y Fo(\015)5 b Fu(\(1\))27 b(=)h(1)k(and)h Fo(\015)5 b Fu(\()p Fo(A)750 4119 y Fw(\003)789 4155 y Fo(A)p Fu(\))28 b Fn(\025)g Fo(\015)5 b Fu(\()p Fo(A)p Fu(\))1238 4119 y Fw(\003)1278 4155 y Fo(\015)g Fu(\()p Fo(A)p Fu(\).)0 4342 y Fj(Theorem)75 b(4.4)38 b(\(Uhlmann's)h (monotonicit)m(y)f(theorem\))49 b Fg(L)-5 b(et)44 b Fo( )2716 4357 y Fl(i)2745 4342 y Fo(;)17 b(\036)2847 4357 y Fl(i)2919 4342 y Fg(b)-5 b(e)43 b(normal)h(states)g(on)0 4462 y Fp(M)105 4477 y Fl(i)133 4462 y Fg(,)34 b Fo(i)28 b Fu(=)g(1)p Fo(;)17 b Fu(2)p Fg(,)34 b(and)g(let)h Fo(\015)e Fu(:)28 b Fp(M)1138 4477 y Fk(1)1204 4462 y Fn(7!)g Fp(M)1437 4477 y Fk(2)1511 4462 y Fg(b)-5 b(e)34 b(a)h(Schwartz)f(map)g(such)h (that)1633 4646 y Fo( )1696 4661 y Fk(2)1758 4646 y Fn(\016)22 b Fo(\015)32 b Fu(=)c Fo( )2080 4661 y Fk(1)2120 4646 y Fo(;)1383 b Fu(\(4.18\))1638 4830 y Fo(\036)1696 4845 y Fk(2)1758 4830 y Fn(\016)22 b Fo(\015)32 b Fu(=)c Fo(\036)2075 4845 y Fk(1)2114 4830 y Fo(:)1389 b Fu(\(4.19\))0 4989 y Fg(Then)1350 5109 y Fu(En)m(t)q(\()p Fo( )1607 5124 y Fk(2)1647 5109 y Fn(j)p Fo(\036)1733 5124 y Fk(2)1772 5109 y Fu(\))27 b Fn(\024)i Fu(En)m(t\()p Fo( )2199 5124 y Fk(1)2239 5109 y Fn(j)p Fo(\036)2325 5124 y Fk(1)2364 5109 y Fu(\))p Fo(:)1841 5753 y Fu(22)p eop %%Page: 23 23 23 22 bop 146 407 a Fu(The)34 b(follo)m(wing)f(inequalit)m(y)h(is)f(a)g (consequence)i(of)d(Uhlmann's)i(theorem:)0 635 y Fj(Corollary)k(4.5)49 b Fg(L)-5 b(et)29 b Fp(N)e Fn(\032)i Fp(M)f Fg(b)-5 b(e)28 b Fo(W)1420 599 y Fw(\003)1459 635 y Fg(-algebr)-5 b(as)28 b(with)g(c)-5 b(ommon)28 b(identity)h(and)f Fo( )t(;)17 b(\036)27 b Fn(2)h Fp(M)3374 599 y Fk(+)p Fl(;)p Fk(1)3374 660 y Fw(\003)3488 635 y Fg(.)42 b(Then)1276 855 y Fu(En)m(t)1432 870 y Fh(M)1508 855 y Fu(\()p Fo( )t Fn(j)p Fo(\036)p Fu(\))27 b Fn(\024)i Fu(En)m(t\()p Fo( )2130 770 y Fc(\014)2130 830 y(\014)2163 894 y Fh(N)2242 855 y Fn(j)17 b Fo(\036)2345 770 y Fc(\014)2345 830 y(\014)2377 894 y Fh(N)2438 855 y Fu(\))p Fo(:)0 1303 y Fj(Pro)s(of.)48 b Fu(The)24 b(inclusion)i(map)e Fo(\015)33 b Fu(:)27 b Fp(N)h Fn(!)f Fp(M)d Fu(is)g(Sc)m(h)m(w)m(artz)i (and)e(satis\014es)i(the)e(conditions)h(of)e(Theorem)0 1424 y(4.4)32 b(with)h(resp)s(ect)h(to)e Fo( )t(;)17 b(\036)32 b Fu(and)h(the)g(restricted)h(states)g Fo( )2170 1339 y Fc(\014)2170 1399 y(\014)2203 1463 y Fh(N)2264 1424 y Fu(,)f Fo(\036)2382 1339 y Fc(\014)2382 1399 y(\014)2415 1463 y Fh(N)2476 1424 y Fu(.)44 b Fe(2)146 1594 y Fu(T)-8 b(o)35 b(pro)m(v)m(e)h(Uhlmann's)h(theorem)e(it)g(is)g(con)m(v)m(enien) m(t)j(to)c(w)m(ork)i(in)f(the)g(standard)h(represen)m(tation)0 1714 y(and)c(to)g(translate)g(the)g(problem)h(in)m(to)f(the)h(language) f(of)f(op)s(erators)h(on)g(Hilb)s(ert)g(spaces.)45 b(Hence)33 b(w)m(e)0 1835 y(assume)d(that)d Fp(M)644 1850 y Fl(i)700 1835 y Fn(\032)h(B)s Fu(\()p Fn(H)995 1850 y Fl(i)1024 1835 y Fu(\))g(and)g(that)g(\()p Fp(M)1625 1850 y Fl(i)1652 1835 y Fo(;)17 b Fn(H)1780 1850 y Fl(i)1809 1835 y Fo(;)g(J)1907 1850 y Fl(i)1934 1835 y Fo(;)g Fn(H)2063 1794 y Fk(+)2062 1860 y Fl(i)2122 1835 y Fu(\))28 b(is)h(a)f(standard)g(form.)42 b(Let)28 b Fo(\015)33 b Fu(:)28 b Fp(M)3442 1850 y Fk(1)3508 1835 y Fn(!)g Fp(M)3741 1850 y Fk(2)0 1955 y Fu(b)s(e)47 b(a)f(Sc)m(h)m(w)m(artz)i(map.)85 b(Let)46 b Fo( )1217 1970 y Fl(i)1297 1955 y Fn(2)51 b Fp(M)1519 1914 y Fk(+)1519 1981 y Fl(i;)p Fw(\003)1648 1955 y Fu(satisfy)d(\(4.18\))d(and)h(let)h (\011)2696 1970 y Fl(i)2770 1955 y Fu(b)s(e)g(the)g(standard)f(v)m (ector)0 2084 y(represen)m(tativ)m(es)35 b(of)c Fo( )837 2099 y Fl(i)866 2084 y Fu(.)43 b(Set)32 b Fn(D)1180 2099 y Fk(1)1247 2084 y Fu(:=)c Fp(M)1483 2099 y Fk(1)1522 2084 y Fu(\011)1598 2099 y Fk(1)1657 2084 y Fu(+)20 b(\()p Fp(M)1896 2099 y Fk(1)1935 2084 y Fu(\011)2011 2099 y Fk(1)2050 2084 y Fu(\))2088 2048 y Fw(?)2147 2084 y Fu(.)43 b(W)-8 b(e)32 b(de\014ne)h(a)f(linear)g(map)g Fo(T)41 b Fu(:)28 b Fn(D)3462 2099 y Fk(1)3529 2084 y Fn(7!)f(H)3740 2099 y Fk(2)0 2205 y Fu(b)m(y)1365 2325 y Fo(T)14 b Fu(\()p Fo(A)p Fu(\011)1623 2340 y Fk(1)1684 2325 y Fu(+)22 b(\002)1858 2340 y Fk(1)1898 2325 y Fu(\))27 b(:=)h Fo(\015)5 b Fu(\()p Fo(A)p Fu(\)\011)2375 2340 y Fk(2)0 2499 y Fu(for)32 b Fo(A)c Fn(2)g Fp(M)449 2514 y Fk(1)520 2499 y Fu(and)33 b(\002)786 2514 y Fk(1)853 2499 y Fn(2)28 b Fu(\()p Fp(M)1090 2514 y Fk(1)1129 2499 y Fu(\011)1205 2514 y Fk(1)1244 2499 y Fu(\))1282 2463 y Fw(?)1341 2499 y Fu(.)44 b(Since)34 b Fo(\015)5 b Fu(\(1\))27 b(=)g(1,)33 b Fo(T)14 b Fu(\011)2234 2514 y Fk(1)2301 2499 y Fu(=)27 b(\011)2480 2514 y Fk(2)2519 2499 y Fu(.)0 2728 y Fj(Lemma)39 b(4.6)49 b Fg(The)34 b(map)h Fo(T)48 b Fg(is)35 b(wel)5 b(l)34 b(de\014ne)-5 b(d)34 b(and)g(extends)g(to)h(a)g(c)-5 b(ontr)g(action)34 b(fr)-5 b(om)35 b Fn(H)3302 2743 y Fk(1)3376 2728 y Fg(to)g Fn(H)3577 2743 y Fk(2)3617 2728 y Fg(.)0 2956 y Fj(Pro)s(of.)1169 3148 y Fn(k)p Fo(\015)5 b Fu(\()p Fo(A)p Fu(\)\011)1500 3163 y Fk(2)1539 3148 y Fn(k)1589 3107 y Fk(2)1656 3148 y Fu(=)28 b Fo( )1823 3163 y Fk(2)1863 3148 y Fu(\()p Fo(\015)5 b Fu(\()p Fo(A)p Fu(\))2106 3107 y Fw(\003)2145 3148 y Fo(\015)g Fu(\()p Fo(A)p Fu(\)\))1656 3364 y Fn(\024)29 b Fo( )1825 3379 y Fk(2)1864 3364 y Fu(\()p Fo(\015)5 b Fu(\()p Fo(A)2069 3323 y Fw(\003)2109 3364 y Fo(A)p Fu(\)\))1656 3580 y(=)28 b Fo( )1823 3595 y Fk(1)1863 3580 y Fu(\()p Fo(A)1974 3539 y Fw(\003)2013 3580 y Fo(A)p Fu(\))g(=)g Fn(k)p Fo(A)p Fu(\011)2455 3595 y Fk(1)2494 3580 y Fn(k)2544 3539 y Fk(2)2583 3580 y Fo(:)3530 3366 y Fu(\(4.20\))0 3799 y(Hence)39 b(if)f(\()p Fo(A)26 b Fn(\000)h Fo(B)5 b Fu(\)\011)824 3814 y Fk(1)900 3799 y Fu(=)37 b(0,)i(then)g(\()p Fo(\015)5 b Fu(\()p Fo(A)p Fu(\))25 b Fn(\000)i Fo(\015)5 b Fu(\()p Fo(B)g Fu(\)\)\011)2053 3814 y Fk(2)2129 3799 y Fu(=)37 b(0.)59 b(Therefore,)41 b Fo(T)52 b Fu(is)38 b(w)m(ell)i(de\014ned.)61 b(By)0 3920 y(\(4.20\),)32 b Fo(T)46 b Fu(is)33 b(a)g(con)m(traction.)44 b Fe(2)146 4090 y Fu(Let)f(\010)401 4105 y Fl(i)472 4090 y Fu(b)s(e)g(the)g(standard)g(v)m(ector)h(represen)m(tativ)m(e)h(of)d Fo(\036)2328 4105 y Fl(i)2356 4090 y Fu(.)73 b(The)44 b(main)f(step)g(of)f(the)h(pro)s(of)f(of)0 4210 y(Theorem)34 b(4.4)e(is)h(the)g(follo)m(wing)g(in)m(terp)s(olation)g(estimate)h(for) e(the)h(relativ)m(e)h(mo)s(dular)e(op)s(erator:)0 4438 y Fj(Lemma)39 b(4.7)49 b Fg(F)-7 b(or)34 b Fu(0)27 b Fn(\024)i Fo(t)f Fn(\024)g Fu(1)p Fg(,)1315 4658 y Fn(k)p Fu(\001)1446 4608 y Fl(t=)p Fk(2)1446 4685 y(\010)1497 4694 y Ff(2)1531 4685 y Fl(;)p Fk(\011)1606 4694 y Ff(2)1645 4658 y Fu(\011)1721 4673 y Fk(2)1760 4658 y Fn(k)f(\024)i(k)p Fu(\001)2074 4608 y Fl(t=)p Fk(2)2074 4685 y(\010)2125 4694 y Ff(1)2159 4685 y Fl(;)p Fk(\011)2234 4694 y Ff(1)2273 4658 y Fu(\011)2349 4673 y Fk(1)2388 4658 y Fn(k)p Fo(:)1841 5753 y Fu(23)p eop %%Page: 24 24 24 23 bop 0 408 a Fj(Pro)s(of.)72 b Fu(The)37 b(space)g Fn(D)920 423 y Fk(1)959 408 y Fu(,)g(de\014ned)h(ab)s(o)m(v)m(e,)g(is)e (a)g(core)g(for)g(\001)2300 357 y Fk(1)p Fl(=)p Fk(2)2300 435 y(\010)2351 444 y Ff(1)2386 435 y Fl(;)p Fk(\011)2461 444 y Ff(1)2499 408 y Fu(.)54 b(Let)36 b Fo(A)e Fn(2)g Fp(M)h Fu(with)i Fo(A)d Fu(=)f Fo(A)p Fu(s)3658 423 y Fk(\011)3713 432 y Ff(1)3752 408 y Fu(.)0 528 y(F)-8 b(or)32 b(\012)245 543 y Fk(1)312 528 y Fu(=)c Fo(A)p Fu(\011)565 543 y Fk(1)626 528 y Fu(+)22 b(\002)800 543 y Fk(1)867 528 y Fn(2)29 b(D)1039 543 y Fk(1)1110 528 y Fu(w)m(e)34 b(get)940 756 y(\001)1021 705 y Fk(1)p Fl(=)p Fk(2)1021 783 y(\010)1072 792 y Ff(2)1106 783 y Fl(;)p Fk(\011)1181 792 y Ff(2)1220 756 y Fo(T)14 b Fu(\012)1361 771 y Fk(1)1428 756 y Fu(=)28 b(\001)1613 705 y Fk(1)p Fl(=)p Fk(2)1613 783 y(\010)1664 792 y Ff(2)1698 783 y Fl(;)p Fk(\011)1773 792 y Ff(2)1812 756 y Fo(\015)5 b Fu(\()p Fo(A)p Fu(\)\011)2093 771 y Fk(2)2160 756 y Fu(=)27 b Fo(J)9 b Fu(s)2364 771 y Fk(\011)2419 780 y Ff(2)2458 756 y Fo(\015)c Fu(\()p Fo(A)p Fu(\))2663 715 y Fw(\003)2703 756 y Fu(\010)2773 771 y Fk(2)2813 756 y Fo(;)1142 1001 y Fu(\001)1223 950 y Fk(1)p Fl(=)p Fk(2)1223 1028 y(\010)1274 1037 y Ff(1)1309 1028 y Fl(;)p Fk(\011)1384 1037 y Ff(1)1422 1001 y Fu(\012)1492 1016 y Fk(1)1560 1001 y Fu(=)27 b(\001)1744 950 y Fk(1)p Fl(=)p Fk(2)1744 1028 y(\010)1795 1037 y Ff(1)1830 1028 y Fl(;)p Fk(\011)1905 1037 y Ff(1)1944 1001 y Fo(A)p Fu(\011)2093 1016 y Fk(1)2160 1001 y Fu(=)g Fo(J)9 b(A)2399 960 y Fw(\003)2439 1001 y Fu(\010)2509 1016 y Fk(1)2549 1001 y Fo(:)0 1224 y Fu(By)33 b(\(4.19\))320 1444 y Fn(k)p Fo(J)9 b Fu(s)471 1459 y Fk(\011)526 1468 y Ff(2)565 1444 y Fo(\015)c Fu(\()p Fo(A)p Fu(\))770 1403 y Fw(\003)809 1444 y Fu(\010)879 1459 y Fk(2)919 1444 y Fn(k)969 1403 y Fk(2)1036 1444 y Fn(\024)28 b Fo(\036)1199 1459 y Fk(2)1238 1444 y Fu(\()p Fo(\015)5 b Fu(\()p Fo(A)p Fu(\))p Fo(\015)g Fu(\()p Fo(A)p Fu(\))1686 1403 y Fw(\003)1726 1444 y Fu(\))27 b Fn(\024)h Fo(\036)1954 1459 y Fk(2)1994 1444 y Fu(\()p Fo(\015)5 b Fu(\()p Fo(AA)2272 1403 y Fw(\003)2311 1444 y Fu(\)\))28 b(=)f Fo(\036)2576 1459 y Fk(1)2615 1444 y Fu(\()p Fo(AA)2799 1403 y Fw(\003)2839 1444 y Fu(\))h(=)f Fn(k)p Fo(J)9 b(A)3194 1403 y Fw(\003)3234 1444 y Fu(\010)3304 1459 y Fk(1)3344 1444 y Fn(k)3394 1403 y Fk(2)3433 1444 y Fo(:)0 1664 y Fu(Hence)1286 1784 y Fn(k)p Fu(\001)1417 1733 y Fk(1)p Fl(=)p Fk(2)1417 1811 y(\010)1468 1820 y Ff(2)1502 1811 y Fl(;)p Fk(\011)1577 1820 y Ff(2)1616 1784 y Fo(T)14 b Fu(\012)1757 1799 y Fk(1)1796 1784 y Fn(k)28 b Fu(=)f Fn(k)p Fu(\001)2108 1733 y Fk(1)p Fl(=)p Fk(2)2108 1811 y(\010)2159 1820 y Ff(1)2194 1811 y Fl(;)p Fk(\011)2269 1820 y Ff(1)2307 1784 y Fu(\012)2377 1799 y Fk(1)2417 1784 y Fn(k)p Fo(:)0 1958 y Fu(By)33 b(Lemma)g(4.6,)g Fo(T)46 b Fu(is)33 b(a)f(con)m(traction.)44 b(Hence,)34 b(b)m(y)g(Theorem)g(A.7,)e(for)g Fo(t)c Fn(2)g Fu([0)p Fo(;)17 b Fu(1])1285 2192 y Fn(k)p Fu(\001)1416 2141 y Fl(t=)p Fk(2)1416 2219 y(\010)1467 2228 y Ff(2)1502 2219 y Fl(;)p Fk(\011)1577 2228 y Ff(2)1615 2192 y Fo(T)d Fu(\012)1756 2207 y Fk(1)1795 2192 y Fn(k)28 b(\024)g(k)p Fu(\001)2109 2141 y Fl(t=)p Fk(2)2109 2219 y(\010)2160 2228 y Ff(1)2195 2219 y Fl(;)p Fk(\011)2270 2228 y Ff(1)2308 2192 y Fu(\012)2378 2207 y Fk(1)2418 2192 y Fn(k)p Fo(:)0 2412 y Fu(Setting)33 b(\012)406 2427 y Fk(1)474 2412 y Fu(=)27 b(\011)653 2427 y Fk(1)725 2412 y Fu(w)m(e)34 b(deriv)m(e)g(the)f(statemen)m(t.)45 b Fe(2)0 2582 y Fj(Pro)s(of)d(of)g(Theorem)g(4.4.)56 b Fu(Using)37 b(Theorem)h(4.3)e (\(1\),)h(Lemma)g(4.7)f(and)h(1)d(=)g Fn(k)p Fu(\011)3225 2597 y Fk(1)3264 2582 y Fn(k)3314 2546 y Fk(2)3388 2582 y Fu(=)g Fn(k)p Fu(\011)3624 2597 y Fk(2)3663 2582 y Fn(k)3713 2546 y Fk(2)3752 2582 y Fu(,)0 2703 y(w)m(e)g(obtain)930 2925 y(En)m(t)q(\()p Fo( )1187 2940 y Fk(2)1226 2925 y Fn(j)p Fo(\036)1312 2940 y Fk(2)1351 2925 y Fu(\))28 b(=)f(lim)1540 2988 y Fl(t)p Fw(#)p Fk(0)1672 2925 y Fo(t)1707 2884 y Fw(\000)p Fk(1)1819 2814 y Fc(\020)1878 2925 y Fn(k)p Fu(\001)2009 2874 y Fl(t=)p Fk(2)2009 2952 y(\010)2060 2961 y Ff(2)2095 2952 y Fl(;)p Fk(\011)2170 2961 y Ff(2)2208 2925 y Fu(\011)2284 2940 y Fk(2)2323 2925 y Fn(k)2373 2884 y Fk(2)2435 2925 y Fn(\000)22 b(k)p Fu(\011)2660 2940 y Fk(2)2699 2925 y Fn(k)2749 2884 y Fk(2)2789 2814 y Fc(\021)1417 3229 y Fn(\024)28 b Fu(lim)1542 3292 y Fl(t)p Fw(#)p Fk(0)1674 3229 y Fo(t)1709 3188 y Fw(\000)p Fk(1)1820 3119 y Fc(\020)1880 3229 y Fn(k)p Fu(\001)2011 3178 y Fl(t=)p Fk(2)2011 3256 y(\010)2062 3265 y Ff(1)2096 3256 y Fl(;)p Fk(\011)2171 3265 y Ff(1)2210 3229 y Fu(\011)2286 3244 y Fk(1)2325 3229 y Fn(k)2375 3188 y Fk(2)2436 3229 y Fn(\000)23 b(k)p Fu(\011)2662 3244 y Fk(1)2701 3229 y Fn(k)2751 3188 y Fk(2)2790 3119 y Fc(\021)1417 3503 y Fu(=)k(En)m(t)q(\()p Fo( )1777 3518 y Fk(1)1817 3503 y Fn(j)p Fo(\036)1903 3518 y Fk(1)1942 3503 y Fu(\))p Fo(:)0 3692 y Fe(2)0 4074 y Ft(5)161 b(P)l(erturbation) 52 b(theory)g(of)i(KMS)g(states)0 4293 y Fu(Let)31 b Fo(\014)i(>)28 b Fu(0.)43 b(In)31 b(this)g(section,)i(giv)m(en)f(a)e (\()p Fo(\034)6 b(;)17 b(\014)6 b Fu(\)-KMS)30 b(state)i Fo(!)i Fu(and)d(a)f(p)s(erturbation)h Fo(Q)p Fu(,)h(w)m(e)g(describ)s (e)0 4414 y(the)k(construction)h(of)e(the)h(p)s(erturb)s(ed)g Fo(\014)6 b Fu(-KMS)36 b(state)g Fo(!)2128 4429 y Fl(Q)2187 4414 y Fu(.)53 b(W)-8 b(e)36 b(also)g(pro)m(v)m(e)g(v)-5 b(arious)36 b(prop)s(erties)h(of)0 4534 y(this)d(state,)h(including)g (the)f(P)m(eierls-Bogoliub)s(o)m(v)h(and)f(the)g(Golden-Thompson)h (inequalities.)49 b(The)0 4654 y(Golden-Thompson)34 b(inequalit)m(y)g (pla)m(ys)g(an)e(imp)s(ortan)m(t)h(role)g(in)g(our)f(construction.)146 4775 y(The)43 b(construction)g(is)f(p)s(erformed)g(on)g(three)g(lev)m (els:)64 b(for)42 b(analytic)g(p)s(erturbations,)j(b)s(ounded)0 4895 y(p)s(erturbations)35 b(and)g(a)f(class)h(of)f(un)m(b)s(ounded)i (p)s(erturbations.)50 b(Although)34 b(the)h(results)h(on)e(the)h (\014rst)0 5015 y(t)m(w)m(o)f(lev)m(els)g(are)f(w)m(ell)h(kno)m(wn,)h (the)e(metho)s(d)g(of)f(the)i(pro)s(of)e(on)g(the)i(second)g(lev)m(el)g (\(b)s(ounded)g(p)s(ertur-)0 5136 y(bations\))i(is)h(new.)55 b(The)37 b(results)g(concerning)g(un)m(b)s(ounded)h(p)s(erturbations)e (are)g(new)h(and)g(they)f(are)0 5256 y(the)d(main)g(results)h(of)e(our) g(pap)s(er.)1841 5753 y(24)p eop %%Page: 25 25 25 24 bop 0 407 a Fs(5.1)135 b(Bounded)44 b(p)t(erturbations)0 592 y Fu(Let)f(\()p Fp(M)p Fo(;)17 b Fn(H)q Fo(;)g(J)n(;)g Fn(H)682 555 y Fk(+)740 592 y Fu(\))42 b(b)s(e)h(a)g Fo(W)1161 555 y Fw(\003)1200 592 y Fu(-algebra)f(in)h(the)g(standard)g (form.)73 b(Let)43 b Fo(\034)54 b Fu(b)s(e)43 b(the)g Fo(W)3309 555 y Fw(\003)3348 592 y Fu(-dynamics)0 712 y(on)e Fp(M)g Fu(with)h(the)g(standard)g(Liouvillian)g Fo(L)p Fu(.)70 b(Let)42 b Fo(!)j Fu(b)s(e)c(a)g(faithful)h(\()p Fo(\034)6 b(;)17 b(\014)6 b Fu(\)-KMS)40 b(state)i(with)g(the)0 832 y(standard)33 b(v)m(ector)h(represen)m(tativ)m(e)h(\012.)146 953 y(Let)i Fo(Q)e Fn(2)f Fp(M)i Fu(b)s(e)h(self-adjoin)m(t)g(and)f Fo(\034)1556 968 y Fl(Q)1653 953 y Fu(the)h(p)s(erturb)s(ed)g Fo(W)2390 917 y Fw(\003)2429 953 y Fu(-dynamics)h(de\014ned)g(b)m(y)f (\(3.6\).)55 b(By)0 1073 y(Theorem)34 b(3.5,)e Fo(L)662 1088 y Fl(Q)750 1073 y Fu(=)27 b Fo(L)c Fu(+)f Fo(Q)g Fn(\000)h Fo(J)9 b(QJ)42 b Fu(is)33 b(the)g(standard)g(Liouvillean)h (of)e Fo(\034)2804 1088 y Fl(Q)2864 1073 y Fu(.)146 1193 y(The)42 b(follo)m(wing)f(t)m(w)m(o)g(theorems)h(summarize)g(the)f(\(b) s(ounded\))g(p)s(erturbation)g(theory)g(of)f(KMS)0 1314 y(states)33 b(dev)m(elop)s(ed)i(b)m(y)e(Araki.)0 1542 y Fj(Theorem)75 b(5.1)90 b Fu(\(1\))49 b(\012)28 b Fn(2)g(D)s Fu(\(e)1272 1506 y Fw(\000)p Fl(\014)s Fk(\()p Fl(L)p Fk(+)p Fl(Q)p Fk(\))p Fl(=)p Fk(2)1658 1542 y Fu(\))p Fg(.)45 b(Set)920 1762 y Fu(\012)990 1777 y Fl(Q)1078 1762 y Fu(:=)27 b(e)1251 1721 y Fw(\000)p Fl(\014)s Fk(\()p Fl(L)p Fk(+)p Fl(Q)p Fk(\))p Fl(=)p Fk(2)1638 1762 y Fu(\012)p Fo(;)156 b(!)1952 1777 y Fl(Q)2012 1762 y Fu(\()p Fo(A)p Fu(\))27 b(=)h(\(\012)2400 1777 y Fl(Q)2460 1762 y Fn(j)p Fo(A)p Fu(\012)2631 1777 y Fl(Q)2691 1762 y Fu(\))p Fo(=)p Fn(k)p Fu(\012)2898 1777 y Fl(Q)2958 1762 y Fn(k)3008 1721 y Fk(2)3047 1762 y Fo(:)41 2024 y Fu(\(2\))48 b(\012)284 2039 y Fl(Q)372 2024 y Fn(2)28 b(H)551 1988 y Fk(+)610 2024 y Fg(.)41 2186 y Fu(\(3\))48 b(\012)284 2201 y Fl(Q)379 2186 y Fg(is)35 b(a)g(cyclic)f(and)h(sep)-5 b(ar)g(ating)34 b(ve)-5 b(ctor)34 b(for)h Fp(M)p Fg(.)41 2347 y Fu(\(4\))48 b Fg(The)35 b(state)g Fo(!)709 2362 y Fl(Q)803 2347 y Fg(is)g(a)f Fu(\()p Fo(\034)1072 2362 y Fl(Q)1132 2347 y Fo(;)17 b(\014)6 b Fu(\))p Fg(-KMS)34 b(state.)41 2509 y Fu(\(5\))48 b(log)18 b(\001)438 2524 y Fk(\012)489 2535 y Fb(Q)573 2509 y Fu(=)28 b Fn(\000)p Fo(\014)6 b(L)881 2524 y Fl(Q)941 2509 y Fg(.)41 2671 y Fu(\(6\))48 b Fg(F)-7 b(or)34 b(al)5 b(l)35 b(self-adjoint)e Fo(Q)1118 2686 y Fk(1)1158 2671 y Fo(;)17 b(Q)1279 2686 y Fk(2)1346 2671 y Fn(2)28 b Fp(M)p Fg(,)1096 2891 y Fu(\(\012)1204 2906 y Fl(Q)1260 2915 y Ff(1)1299 2891 y Fu(\))1337 2906 y Fl(Q)1393 2915 y Ff(2)1458 2891 y Fu(=)g(\012)1632 2906 y Fl(Q)1688 2915 y Ff(1)1723 2906 y Fk(+)p Fl(Q)1834 2915 y Ff(2)1872 2891 y Fo(;)216 b Fu(\()p Fo(!)2214 2906 y Fl(Q)2270 2915 y Ff(1)2308 2891 y Fu(\))2346 2906 y Fl(Q)2402 2915 y Ff(2)2467 2891 y Fu(=)28 b Fo(!)2632 2906 y Fl(Q)2688 2915 y Ff(1)2722 2906 y Fk(+)p Fl(Q)2833 2915 y Ff(2)2871 2891 y Fo(:)41 3153 y Fu(\(7\))48 b(log)18 b(\001)438 3168 y Fk(\012)489 3179 y Fb(Q)542 3168 y Fl(;)p Fk(\012)644 3153 y Fu(=)28 b(log)17 b(\001)971 3168 y Fk(\012)1049 3153 y Fn(\000)22 b Fo(\014)6 b(Q)p Fg(.)41 3315 y Fu(\(8\))48 b(log)18 b(\001)438 3330 y Fk(\012)p Fl(;)p Fk(\012)560 3341 y Fb(Q)644 3315 y Fu(=)28 b(log)17 b(\001)971 3330 y Fk(\012)1022 3341 y Fb(Q)1101 3315 y Fu(+)22 b Fo(\014)6 b(Q)p Fg(.)41 3477 y Fu(\(9\))48 b(En)m(t)q(\()p Fo(!)t Fn(j)p Fo(!)562 3492 y Fl(Q)621 3477 y Fu(\))22 b(+)g Fo(\014)6 b(!)t Fu(\()p Fo(Q)p Fu(\))26 b(=)i Fn(\000)17 b Fu(log)h Fn(k)p Fu(\012)1545 3492 y Fl(Q)1605 3477 y Fn(k)1655 3440 y Fk(2)1694 3477 y Fg(.)-6 3638 y Fu(\(10\))48 b(En)m(t)q(\()p Fo(!)471 3653 y Fl(Q)530 3638 y Fn(j)p Fo(!)t Fu(\))22 b Fn(\000)g Fo(\014)6 b(!)904 3653 y Fl(Q)963 3638 y Fu(\()p Fo(Q)p Fu(\))28 b(=)f(log)18 b Fn(k)p Fu(\012)1510 3653 y Fl(Q)1570 3638 y Fn(k)1620 3602 y Fk(2)1659 3638 y Fg(.)-6 3800 y Fu(\(11\))48 b Fg(The)35 b(Peierls-Bo)-5 b(goliub)g(ov)32 b(ine)-5 b(quality)35 b(holds:)1578 4020 y Fu(e)1621 3979 y Fw(\000)p Fl(\014)s Fk(\(\012)p Fw(j)p Fl(Q)p Fk(\012\))p Fl(=)p Fk(2)2054 4020 y Fn(\024)28 b(k)p Fu(\012)2279 4035 y Fl(Q)2339 4020 y Fn(k)p Fo(:)-6 4282 y Fu(\(12\))48 b Fg(The)35 b(Golden-Thompson)d(ine)-5 b(quality)35 b(holds:)1581 4502 y Fn(k)p Fu(\012)1701 4517 y Fl(Q)1761 4502 y Fn(k)28 b(\024)g(k)p Fu(e)2037 4461 y Fw(\000)p Fl(\014)s(Q=)p Fk(2)2265 4502 y Fu(\012)p Fn(k)p Fo(:)-6 4763 y Fu(\(13\))48 b Fg(Assume)38 b(that)f Fo(Q)861 4778 y Fl(n)941 4763 y Fn(2)32 b Fp(M)37 b Fg(ar)-5 b(e)37 b(self-adjoint)g(and)f Fo(Q)2126 4778 y Fl(n)2206 4763 y Fn(!)c Fo(Q)37 b Fg(str)-5 b(ongly.)52 b(Then)37 b Fu(\012)3191 4778 y Fl(Q)3247 4786 y Fb(n)3326 4763 y Fn(!)32 b Fu(\012)3528 4778 y Fl(Q)3625 4763 y Fg(and)214 4884 y Fo(!)275 4899 y Fl(Q)331 4907 y Fb(n)405 4884 y Fn(!)27 b Fo(!)593 4899 y Fl(Q)688 4884 y Fg(in)34 b(norm.)1841 5753 y Fu(25)p eop %%Page: 26 26 26 25 bop 0 407 a Fj(Theorem)75 b(5.2)49 b Fg(L)-5 b(et)348 621 y Fo(T)405 636 y Fl(\014)s(;n)543 621 y Fu(=)27 b Fn(f)p Fu(\()p Fo(\014)789 636 y Fk(1)829 621 y Fo(;)17 b Fn(\001)g(\001)g(\001)31 b Fo(;)17 b(\014)1121 636 y Fl(n)1167 621 y Fu(\))28 b Fn(2)g Fm(R)1399 580 y Fl(n)1509 621 y Fu(:)62 b Fo(\014)1653 636 y Fl(i)1709 621 y Fn(\025)28 b Fu(0)p Fo(;)52 b(i)28 b Fu(=)f(1)p Fo(;)17 b(:)g(:)g(:)f(;)h(n;)51 b(\014)2565 636 y Fk(1)2627 621 y Fu(+)22 b Fn(\001)17 b(\001)g(\001)j Fu(+)i Fo(\014)3016 636 y Fl(n)3091 621 y 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1563 y Fl(;)p Fw(\001\001\001)10 b Fl(;\014)1222 1571 y Fb(n)1264 1563 y Fk(\))p Fw(2)p Fl(T)1379 1575 y Fb(\014)s(;n)1478 1477 y Fn(k)p Fu(e)1571 1436 y Fw(\000)p Fl(\014)1666 1445 y Ff(1)1700 1436 y Fl(L)1752 1477 y Fo(Q)17 b Fn(\001)g(\001)g(\001)e Fu(e)2022 1436 y Fw(\000)p Fl(\014)2117 1444 y Fb(n)2159 1436 y Fl(L)2211 1477 y Fo(Q)i Fu(\012)p Fn(k)28 b(\024)g(k)p Fo(Q)p Fn(k)2735 1436 y Fl(n)2782 1477 y Fo(;)721 b Fu(\(5.21\))0 1766 y Fg(and)732 1937 y Fu(\012)802 1952 y Fl(Q)890 1937 y Fu(=)1030 1813 y Fw(1)993 1842 y Fc(X)999 2052 y Fl(n)p Fk(=0)1137 1937 y Fu(\()p Fn(\000)p Fu(1\))1339 1896 y Fl(n)1386 1802 y Fc(Z)1486 1937 y Fn(\001)17 b(\001)g(\001)1602 1802 y Fc(Z)1474 2085 y Fl(T)1515 2097 y Fb(\014)s(;n)1702 1937 y Fu(e)1745 1896 y Fw(\000)p Fl(\014)1840 1905 y Ff(1)1874 1896 y Fl(L)1926 1937 y Fo(Q)g Fn(\001)g(\001)g(\001)e Fu(e)2196 1896 y Fw(\000)p Fl(\014)2291 1904 y Fb(n)2333 1896 y Fl(L)2385 1937 y Fo(Q)i Fu(\012)g(d)p Fo(\014)2675 1952 y Fk(1)2732 1937 y Fn(\001)g(\001)g(\001)d Fu(d)p Fo(\014)2973 1952 y Fl(n)3021 1937 y Fo(:)482 b Fu(\(5.22\))146 2494 y(W)-8 b(e)39 b(ha)m(v)m(e)h(separated)g(Theorem)g(5.2)e(from)g (the)h(other)g(results)h(of)e(Araki's)i(theory)f(for)f(sev)m(eral)0 2614 y(reasons.)146 2735 y(Theorem)33 b(5.2)e(con)m(tains)h(the)g(main) f(idea)h(of)f(Araki's)h(original)f(pro)s(of)f(of)h(Theorem)h(5.1.)43 b(In)32 b(fact,)0 2855 y(his)j(pro)s(of)f(w)m(as)h(cen)m(tered)i (around)d(the)h(expansion)h(\(5.22\).)49 b(Our)35 b(metho)s(ds)g(are)g (in)f(a)h(certain)g(sense)0 2975 y(orthogonal)d(to)g(Araki's)h(and)g(w) m(e)h(do)e(not)h(need)g(Theorem)h(5.2)e(to)h(pro)m(v)m(e)g(Theorem)h (5.1.)146 3096 y(The)g(expansion)f(\(5.22\))f(is)h(an)f(additional)g (information)h(ab)s(out)e(\012)2651 3111 y Fl(Q)2744 3096 y Fu(whic)m(h,)j(strictly)f(sp)s(eaking,)0 3216 y(cannot)h(b)s(e)g(deriv)m(ed)i(b)m(y)f(our)f(metho)s(ds)h(alone.)48 b(Hence,)35 b(for)f(b)s(ounded)h(p)s(erturbations)f(our)g(metho)s(d)0 3337 y(yields)f(a)d(sligh)m(tly)j(w)m(eak)m(er)g(result)f(than)f(the)g (Araki)h(metho)s(d.)43 b(On)31 b(the)h(other)f(hand,)h(our)f(metho)s(d) g(is)0 3457 y(simpler)j(and)f(easily)g(extends)i(to)d(a)g(large)h (class)h(of)e(un)m(b)s(ounded)i(p)s(erturbations)f Fo(Q)p Fu(.)146 3577 y(Both)47 b(Araki)h(and)f(our)g(metho)s(ds)h(start)f (with)h(analytic)f(p)s(erturbations.)88 b(In)48 b(this)f(case,)52 b(the)0 3698 y(pro)s(ofs)29 b(of)g(Theorems)j(5.1)d(and)h(5.2)f(are)g (essen)m(tially)k(algebraic)c(and)h(relativ)m(ely)i(easy)-8 b(.)43 b(F)-8 b(or)29 b(a)g(general)0 3818 y(b)s(ounded)35 b Fo(Q)f Fu(one)h(pic)m(ks)h(a)e(sequence)j(of)c(analytic)i Fo(Q)1989 3833 y Fl(n)2070 3818 y Fu(with)g Fo(Q)2371 3833 y Fl(n)2449 3818 y Fn(!)30 b Fo(Q)k Fu(and)h(uses)g(v)-5 b(arious)35 b(limit)f(ar-)0 3938 y(gumen)m(ts)29 b(to)f(establish)h (the)g(theorems.)43 b(The)29 b(k)m(ey)g(di\013erence)h(b)s(et)m(w)m (een)g(the)e(t)m(w)m(o)h(metho)s(ds)g(concerns)0 4059 y(these)43 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Fo(\014)6 b Fu(\(\012)2451 878 y Fl(Q)2510 863 y Fn(j)p Fo(Q)p Fu(\012)2685 878 y Fl(Q)2745 863 y Fu(\))p Fo(=)p Fu(2)p Fo(:)41 1079 y Fu(\(5\))1261 1199 y Fn(k)p Fu(\012)1381 1214 y Fl(Q)1463 1199 y Fn(\000)22 b Fu(\012)p Fn(k)28 b(\024)g Fo(\014)6 b(f)11 b Fu(\()p Fn(k)p Fo(Q)p Fu(\012)p Fn(k)p Fo(;)17 b Fn(k)p Fo(Q)p Fu(\012)2461 1214 y Fl(Q)2521 1199 y Fn(k)p Fu(\))p Fo(=)p Fu(2)p Fo(;)214 1373 y Fg(wher)-5 b(e,)35 b(for)f Fo(x;)17 b(y)31 b(>)d Fu(0)p Fg(,)34 b(we)h(set)1322 1681 y Fo(f)11 b Fu(\()p Fo(x;)17 b(y)t Fu(\))27 b(:=)1766 1511 y Fc(\()2000 1541 y Fl(x)p Fw(\000)p Fl(y)p 1897 1562 338 4 v 1897 1619 a Fk(log)13 b Fl(x)p Fw(\000)p Fk(log)g Fl(y)2245 1585 y Fo(;)83 b(x)28 b Fn(6)p Fu(=)g Fo(y)t Fu(;)2052 1776 y Fo(x)248 b(x)28 b Fu(=)g Fo(y)t(:)146 2102 y Fu(The)40 b(estimate)g(\(1\))e(follo)m(ws) i(immediately)h(from)d(\(5.21\))g(and)h(is)g(of)f(course)i(w)m(ell-kno) m(wn.)65 b(The)0 2222 y(estimates)34 b(\(2\){\(5\))e(app)s(ear)g(to)g (b)s(e)h(new.)0 2511 y Fs(5.2)135 b(Analytic)46 b(p)t(erturbations|pro) t(ofs)0 2696 y Fu(In)28 b(this)h(section)g(w)m(e)g(pro)m(v)m(e)h (Theorem)f(5.1)f(for)f(analytic)i(self-adjoin)m(t)f(p)s(erturbations)h Fo(Q)f Fn(2)g Fp(M)3500 2711 y Fl(\034)3543 2696 y Fu(.)42 b(The)0 2816 y(pro)s(ofs)32 b(are)h(based)g(on)g(the)g(algebraic)g (argumen)m(ts)g(and)g(are)g(relativ)m(ely)h(easy)-8 b(.)0 2936 y Fj(Pro)s(of)37 b(of)h(Theorem)g(5.1)f(in)h(the)g(analytic)g (case)33 b Fu(\(1\))f(F)-8 b(or)32 b Fo(t)h Fu(real,)1117 3156 y(E)1183 3171 y Fl(Q)1243 3156 y Fu(\()p Fo(t)p Fu(\)\012)28 b(=)g(e)1599 3115 y Fk(i)p Fl(t)p Fk(\()p Fl(L)p Fk(+)p Fl(Q)p Fk(\))1862 3156 y Fu(e)1905 3115 y Fw(\000)p Fk(i)p Fl(tL)2057 3156 y Fu(\012)g(=)g(e)2302 3115 y Fk(i)p Fl(t)p Fk(\()p Fl(L)p Fk(+)p Fl(Q)p Fk(\))2565 3156 y Fu(\012)p Fo(:)0 3376 y Fu(Since)e(E)313 3391 y Fl(Q)373 3376 y Fu(\()p Fo(t)p Fu(\))g(has)f(an)g(analytic)h(con)m (tin)m(uation)h(to)d(an)i(en)m(tire)g(function)g Fo(z)32 b Fn(7!)27 b Fu(E)2876 3391 y Fl(Q)2936 3376 y Fu(\()p Fo(z)t Fu(\),)h(\012)g Fn(2)g(D)s Fu(\(e)3469 3340 y Fk(i)p Fl(z)s Fk(\()p Fl(L)p Fk(+)p Fl(Q)p Fk(\))3742 3376 y Fu(\))0 3496 y(for)k(all)h Fo(z)f Fn(2)c Fm(C)33 b Fu(and)f(E)816 3511 y Fl(Q)876 3496 y Fu(\()p Fo(z)t Fu(\)\012)d(=)f(e)1247 3460 y Fk(i)p Fl(z)s Fk(\()p Fl(L)p Fk(+)p Fl(Q)p Fk(\))1520 3496 y Fu(\012)p Fo(:)33 b Fu(In)f (particular,)1517 3716 y(\012)1587 3731 y Fl(Q)1675 3716 y Fu(=)27 b(E)1844 3731 y Fl(Q)1904 3716 y Fu(\(i)p Fo(\014)6 b(=)p Fu(2\)\012)p Fo(:)1267 b Fu(\(5.23\))146 3935 y(\(2\))32 b(W)-8 b(e)33 b(ha)m(v)m(e)1077 4159 y(E)1143 4174 y Fl(Q)1203 4159 y Fu(\(i)p Fo(\014)6 b(=)p Fu(2\))27 b(=)g(E)1661 4174 y Fl(Q)1721 4159 y Fu(\(i)p Fo(\014)6 b(=)p Fu(4\))p Fo(\034)2036 4118 y Fk(i)p Fl(\014)s(=)p Fk(4)2173 4159 y Fu(\(E)2277 4174 y Fl(Q)2337 4159 y Fu(\(i)p Fo(\014)g(=)p Fu(4\)\))1492 4390 y(=)27 b(E)1661 4405 y Fl(Q)1721 4390 y Fu(\(i)p Fo(\014)6 b(=)p Fu(4\))p Fo(\034)2036 4349 y Fk(i)p Fl(\014)s(=)p Fk(2)2173 4390 y Fu(\(E)2277 4405 y Fl(Q)2337 4390 y Fu(\(i)p Fo(\014)g(=)p Fu(4\))2599 4349 y Fw(\003)2638 4390 y Fu(\))p Fo(:)0 4608 y Fu(Hence,)34 b(b)m(y)f(\(5.23\),)1185 4832 y(\012)1255 4847 y Fl(Q)1343 4832 y Fu(=)27 b(E)1512 4847 y Fl(Q)1572 4832 y Fu(\(i)p Fo(\014)6 b(=)p Fu(4\)e)1877 4791 y Fw(\000)p Fl(\014)s(L=)p Fk(2)2097 4832 y Fu(E)2163 4847 y Fl(Q)2223 4832 y Fu(\(i)p Fo(\014)g(=)p Fu(4\))2485 4791 y Fw(\003)2524 4832 y Fu(\012)1343 5048 y(=)27 b(E)1512 5063 y Fl(Q)1572 5048 y Fu(\(i)p Fo(\014)6 b(=)p Fu(4\))p Fo(J)j Fu(E)1963 5063 y Fl(Q)2023 5048 y Fu(\(i)p Fo(\014)d(=)p Fu(4\)\012)p Fo(:)0 5268 y Fu(Therefore,)34 b(\012)539 5283 y Fl(Q)627 5268 y Fn(2)28 b(H)806 5232 y Fk(+)865 5268 y Fu(.)1841 5753 y(27)p eop %%Page: 28 28 28 27 bop 146 407 a Fu(\(3\))36 b(Since)h(E)631 422 y Fl(Q)691 407 y Fu(\(i)p Fo(\014)6 b(=)p Fu(2\))35 b(is)i(an)f(in)m(v)m (ertible)i(elemen)m(t)g(of)e Fp(M)p Fu(,)g(\012)2380 422 y Fl(Q)2476 407 y Fu(is)h(ob)m(viously)h(a)d(cyclic)j(and)e(sepa-)0 527 y(rating)c(v)m(ector)i(for)e Fp(M)p Fu(.)146 648 y(\(4\))g(Theorem)i(3.6)f(yields)946 868 y(e)989 826 y Fw(\000)p Fl(\014)s(L)1135 837 y Fb(Q)1187 826 y Fl(=)p Fk(2)1289 868 y Fu(=)28 b Fo(J)9 b Fu(E)1522 883 y Fl(Q)1582 868 y Fu(\()p Fn(\000)p Fu(i)p Fo(\014)d(=)p Fu(2\))p Fo(J)j Fu(e)2027 826 y Fw(\000)p Fl(\014)s(L=)p Fk(2)2248 868 y Fu(E)2314 883 y Fl(Q)2374 868 y Fu(\()p Fn(\000)p Fu(i)p Fo(\014)d(=)p Fu(2\))2713 826 y Fw(\000)p Fk(1)2807 868 y Fo(;)0 1088 y Fu(and)33 b Fp(M)p Fu(\012)365 1103 y Fl(Q)452 1088 y Fu(=)28 b Fp(M)p Fu(\012)f Fn(\032)h(D)s Fu(\(e)1024 1051 y Fw(\000)p Fl(\014)s(L)1170 1062 y Fb(Q)1223 1051 y Fl(=)p Fk(2)1297 1088 y Fu(\).)44 b(Moreo)m(v)m(er,)34 b(for)e Fo(A)c Fn(2)g Fp(M)p Fu(,)504 1314 y(e)547 1273 y Fw(\000)p Fl(\014)s(L)693 1284 y Fb(Q)745 1273 y Fl(=)p Fk(2)820 1314 y Fo(A)p Fu(\012)963 1329 y Fl(Q)1051 1314 y Fu(=)g Fo(J)9 b Fu(E)1284 1329 y Fl(Q)1344 1314 y Fu(\()p Fn(\000)p Fu(i)p Fo(\014)d(=)p Fu(2\))p Fo(J)j Fu(e)1789 1273 y Fw(\000)p Fl(\014)s(L=)p Fk(2)2009 1314 y Fu(E)2075 1329 y Fl(Q)2136 1314 y Fu(\()p Fn(\000)p Fu(i)p Fo(\014)d(=)p Fu(2\))2475 1273 y Fw(\000)p Fk(1)2568 1314 y Fo(A)p Fu(E)2707 1329 y Fl(Q)2768 1314 y Fu(\(i)p Fo(\014)g(=)p Fu(2\)\012)1051 1530 y(=)28 b Fo(J)9 b Fu(E)1284 1545 y Fl(Q)1344 1530 y Fu(\()p Fn(\000)p Fu(i)p Fo(\014)d(=)p Fu(2\)E)1749 1545 y Fl(Q)1808 1530 y Fu(\(i)p Fo(\014)g(=)p Fu(2\))2070 1489 y Fw(\003)2109 1530 y Fo(A)2182 1489 y Fw(\003)2222 1530 y Fu(E)2288 1545 y Fl(Q)2348 1530 y Fu(\()p Fn(\000)p Fu(i)p Fo(\014)g(=)p Fu(2\))2687 1489 y Fw(\000)p Fk(1)p Fw(\003)2816 1530 y Fu(\012)1051 1747 y(=)28 b Fo(J)9 b Fu(E)1284 1762 y Fl(Q)1344 1747 y Fu(\()p Fn(\000)p Fu(i)p Fo(\014)d(=)p Fu(2\)E)1749 1762 y Fl(Q)1808 1747 y Fu(\()p Fn(\000)p Fu(i)p Fo(\014)g(=)p Fu(2\))2147 1705 y Fw(\000)p Fk(1)2241 1747 y Fo(A)2314 1705 y Fw(\003)2354 1747 y Fu(E)2420 1762 y Fl(Q)2480 1747 y Fu(\(i)p Fo(\014)g(=)p Fu(2\)\012)27 b(=)h Fo(J)9 b(A)3079 1705 y Fw(\003)3118 1747 y Fu(\012)3188 1762 y Fl(Q)3249 1747 y Fo(:)146 1965 y Fu(\(5\))34 b(By)h(Theorem)h(3.5,)e (w)m(e)i(kno)m(w)f(that)f Fo(L)1741 1980 y Fl(Q)1832 1965 y Fu(:=)d Fo(L)24 b Fu(+)f Fo(Q)g Fn(\000)h Fo(J)9 b(QJ)44 b Fu(is)35 b(the)g(Liouvillean)g(of)f Fo(\034)3523 1980 y Fl(Q)3583 1965 y Fu(.)49 b(By)0 2086 y(Theorem)34 b(2.18)e(w)m(e)h(kno)m(w)h(that)f(\001)1309 2101 y Fk(\012)1360 2112 y Fb(Q)1444 2086 y Fu(=)27 b(e)1590 2050 y Fw(\000)p Fl(\014)s(L)1736 2061 y Fb(Q)1793 2086 y Fu(.)146 2206 y(\(6\))32 b(follo)m(ws)i(from)1091 2327 y(E)1157 2266 y Fl(\034)1188 2277 y Fb(Q)1236 2292 y Ff(1)1157 2354 y Fl(Q)1213 2363 y Ff(2)1280 2327 y Fu(\(i)p Fo(\014)6 b(=)p Fu(2\)E)1608 2285 y Fl(\034)1608 2351 y(Q)1664 2360 y Ff(1)1702 2327 y Fu(\(i)p Fo(\014)g(=)p Fu(2\))27 b(=)g(E)2160 2285 y Fl(\034)2160 2351 y(Q)2216 2360 y Ff(1)2251 2351 y Fk(+)p Fl(Q)2362 2360 y Ff(2)2400 2327 y Fu(\(i)p Fo(\014)6 b(=)p Fu(2\))p Fo(;)0 2501 y Fu(whic)m(h)33 b(is)e(an)g(immediate)h(consequence)i(of)d(Theorem)h(3.1)f(\(6\),)g (where)h Fo(L)20 b Fu(+)e Fo(Q)2928 2516 y Fk(1)2999 2501 y Fu(is)32 b(to)e(b)s(e)h(used)i(for)d Fo(L)0 2621 y Fu(in)j(the)g(expression)i(for)d(E)967 2561 y Fl(\034)998 2572 y Fb(Q)1046 2587 y Ff(1)967 2648 y Fl(Q)1023 2657 y Ff(2)1089 2621 y Fu(\()p Fo(t)p Fu(\).)146 2742 y(\(7\))g(The)i (relation)1104 2862 y Fo(S)1164 2877 y Fk(\012)1219 2862 y Fu(E)1285 2877 y Fl(Q)1345 2862 y Fu(\(i)p Fo(\014)6 b(=)p Fu(2\))1607 2821 y Fw(\003)1645 2862 y Fo(A)p Fu(\012)29 b(=)e Fo(A)1993 2821 y Fw(\003)2033 2862 y Fu(\012)2103 2877 y Fl(Q)2191 2862 y Fu(=)g Fo(S)2354 2877 y Fk(\012)2405 2888 y Fb(Q)2457 2877 y Fl(;)p Fk(\012)2532 2862 y Fo(A)p Fu(\012)0 3036 y(implies)34 b(that)1418 3157 y Fo(S)1478 3172 y Fk(\012)1529 3183 y Fb(Q)1581 3172 y Fl(;)p Fk(\012)1684 3157 y Fu(=)27 b Fo(S)1847 3172 y Fk(\012)1902 3157 y Fo(E)1974 3172 y Fl(Q)2034 3157 y Fu(\(i)p Fo(\014)6 b(=)p Fu(2\))2296 3116 y Fw(\003)2335 3157 y Fo(:)0 3331 y Fu(Hence)924 3523 y(\001)1005 3538 y Fk(\012)p Fl(;)p Fk(\012)1127 3549 y Fb(Q)1211 3523 y Fu(=)28 b Fo(S)1381 3482 y Fw(\003)1375 3548 y Fk(\012)1426 3559 y Fb(Q)1478 3548 y Fl(;)p Fk(\012)1553 3523 y Fo(S)1613 3538 y Fk(\012)1664 3549 y Fb(Q)1716 3538 y Fl(;)p Fk(\012)1211 3757 y Fu(=)g(E)1381 3772 y Fl(Q)1441 3757 y Fu(\(i)p Fo(\014)6 b(=)p Fu(2\)\001)1784 3772 y Fk(\012)1838 3757 y Fu(E)1904 3716 y Fw(\003)1904 3782 y Fl(Q)1964 3757 y Fu(\(i)p Fo(\014)g(=)p Fu(2\))1211 3990 y(=)1315 3909 y Fc(\000)1360 3990 y Fu(E)1426 4005 y Fl(Q)1486 3990 y Fu(\(i)p Fo(\014)g(=)p Fu(2\)e)1791 3949 y Fw(\000)p Fl(\014)s(L=)p Fk(2)2011 3909 y Fc(\001)17 b(\000)2119 3990 y Fu(e)2162 3949 y Fw(\000)p Fl(\014)s(L=)p Fk(2)2383 3990 y Fu(E)2449 4005 y Fl(Q)2509 3990 y Fu(\(i)p Fo(\014)6 b(=)p Fu(2\))2771 3949 y Fw(\003)2810 3909 y Fc(\001)1211 4221 y Fu(=)28 b(e)1358 4180 y Fw(\000)p Fl(\014)s Fk(\()p Fl(L)p Fk(+)p Fl(Q)p Fk(\))1673 4221 y Fo(;)0 4447 y Fu(where)34 b(w)m(e)f(used)h(\001)729 4462 y Fk(\012)812 4447 y Fu(=)28 b(e)959 4411 y Fw(\000)p Fl(\014)s(L)1109 4447 y Fu(.)146 4568 y(\(8\))k(follo)m(ws)i(from)e (\(7\))g(if)h(w)m(e)g(note)g(that,)g(b)m(y)h(\(6\),)f(\(\012)2129 4583 y Fl(Q)2189 4568 y Fu(\))2227 4583 y Fw(\000)p Fl(Q)2369 4568 y Fu(=)27 b(\012.)146 4688 y(\(9\))35 b(Set)499 4663 y(~)476 4688 y Fo(Q)e Fu(:=)e Fo(Q)25 b Fu(+)e Fo(\014)982 4652 y Fw(\000)p Fk(1)1093 4688 y Fu(log)17 b Fn(k)p Fu(\012)1355 4703 y Fl(Q)1415 4688 y Fn(k)1465 4652 y Fk(2)1505 4688 y Fu(.)50 b(Then)37 b Fo(!)1901 4703 y Fl(Q)1992 4688 y Fu(=)32 b Fo(!)2177 4698 y Fk(~)2161 4715 y Fl(Q)2255 4688 y Fu(and)j(\012)2533 4698 y Fk(~)2517 4715 y Fl(Q)2609 4688 y Fu(:=)d(\012)2814 4703 y Fl(Q)2874 4688 y Fo(=)p Fn(k)p Fu(\012)3043 4703 y Fl(Q)3103 4688 y Fn(k)p Fu(.)51 b(Using)37 b(\(7\))d(w)m(e)0 4808 y(get)1340 4929 y(log)18 b(\001)1564 4944 y Fk(\012)1629 4955 y Ff(~)1615 4968 y Fb(Q)1668 4944 y Fl(;)p Fk(\012)1770 4929 y Fu(=)27 b(log)18 b(\001)2097 4944 y Fk(\012)2175 4929 y Fn(\000)k Fo(\014)2357 4903 y Fu(~)2335 4929 y Fo(Q;)0 5103 y Fu(whic)m(h)34 b(implies)1411 5223 y(En)m(t)q(\()p Fo(!)t Fn(j)p Fo(!)1759 5238 y Fl(Q)1817 5223 y Fu(\))28 b(=)f Fn(\000)p Fo(\014)6 b(!)t Fu(\()2249 5198 y(~)2227 5223 y Fo(Q)p Fu(\))p Fo(:)1841 5753 y Fu(28)p eop %%Page: 29 29 29 28 bop 146 407 a Fu(\(10\))32 b(Similarly)-8 b(,)34 b(using)g(\(8\))f(w)m(e)g(get)1315 627 y(log)18 b(\001)1539 654 y Fk(\012)p Fl(;)1618 637 y Fk(~)1610 654 y(\012)1661 665 y Fb(Q)1745 627 y Fu(=)27 b(log)18 b(\001)2072 642 y Fk(\012)2137 653 y Ff(~)2123 666 y Fb(Q)2202 627 y Fu(+)k Fo(\014)2383 602 y Fu(~)2361 627 y Fo(Q)o(;)0 847 y Fu(whic)m(h)34 b(implies)1422 967 y(En)m(t\()p Fo(!)1676 982 y Fl(Q)1736 967 y Fn(j)p Fo(!)t Fu(\))26 b(=)i Fo(\014)6 b(!)2119 982 y Fl(Q)2178 967 y Fu(\()2238 942 y(~)2216 967 y Fo(Q)p Fu(\))p Fo(:)146 1142 y Fu(\(11\))32 b(Since)i(En)m(t)q(\()p Fo(!)t Fn(j)p Fo(!)955 1157 y Fl(Q)1013 1142 y Fu(\))28 b Fn(\024)g Fu(0,)k(\(9\))g(yields)j(that) 1471 1362 y(e)1514 1320 y Fw(\000)p Fl(\014)s Fk(\(\012)p Fw(j)p Fl(Q)p Fk(\012\))p Fl(=)p Fk(2)1947 1362 y Fn(\024)28 b(k)p Fu(\012)2172 1377 y Fl(Q)2232 1362 y Fn(k)p Fo(:)0 1582 y Fu(This)34 b(is)f(the)g(P)m(eierls-Bogoliub)s(o)m(v)h(inequalit) m(y)-8 b(.)146 1702 y(\(12\))32 b(Let)h Fp(N)f Fu(b)s(e)h(the)g(Ab)s (elian)g(v)m(on)h(Neumann)f(subalgebra)g(of)g Fp(M)f Fu(generated)h(b)m(y)h Fo(Q)p Fu(.)43 b(Then,)1048 1919 y(log)18 b Fn(k)p Fu(\012)1311 1934 y Fl(Q)1371 1919 y Fn(k)1421 1878 y Fk(2)1488 1919 y Fu(=)27 b(En)m(t)q(\()p Fo(!)1846 1934 y Fl(Q)1906 1919 y Fn(j)p Fo(!)t Fu(\))21 b Fn(\000)h Fo(\014)6 b(!)2279 1934 y Fl(Q)2338 1919 y Fu(\()p Fo(Q)p Fu(\))1488 2144 y Fn(\024)28 b Fu(En)m(t)1749 2063 y Fc(\000)1794 2144 y Fo(!)1855 2159 y Fl(Q)1915 2059 y Fc(\014)1915 2119 y(\014)1948 2183 y Fh(N)2026 2144 y Fn(j)17 b Fo(!)2136 2059 y Fc(\014)2136 2119 y(\014)2168 2183 y Fh(N)2230 2063 y Fc(\001)2297 2144 y Fn(\000)23 b Fo(\014)6 b(!)2519 2159 y Fl(Q)2578 2144 y Fu(\()p Fo(Q)p Fu(\))1488 2374 y Fn(\024)28 b Fu(log)18 b Fo(!)t Fu(\(e)1882 2333 y Fw(\000)p Fl(\014)s(Q)2039 2374 y Fu(\))1488 2605 y(=)27 b(log)18 b Fn(k)p Fu(e)1827 2564 y Fw(\000)p Fl(\014)s(Q=)p Fk(2)2055 2605 y Fu(\012)p Fn(k)2175 2564 y Fk(2)2215 2605 y Fo(;)3530 2264 y Fu(\(5.24\))0 2819 y(and)33 b(so)1474 2939 y Fn(k)p Fu(\012)1594 2954 y Fl(Q)1654 2939 y Fn(k)28 b(\024)g(k)p Fu(e)1930 2898 y Fw(\000)p Fl(\014)s(Q=)p Fk(2)2158 2939 y Fu(\012)p Fn(k)p Fo(:)0 3113 y Fu(This)37 b(is)f(the)f(Golden-Thompson)i (inequalit)m(y)-8 b(.)54 b(In)36 b(the)g(\014rst)g(step)g(of)f (\(5.24\))g(w)m(e)h(used)h(\(10\),)e(in)h(the)0 3234 y(second|Uhlmann's)g(estimate)f(of)e(Corollary)h(4.5)f(and)h(in)g(the)g (third|Theorem)h(4.3)e(\(4\))g(with)h Fo(Q)0 3354 y Fu(replaced)g(b)m (y)f Fn(\000)p Fo(\014)6 b(Q)p Fu(.)146 3475 y(\(13\))24 b(is)h(a)f(general)h(fact)f(whic)m(h)i(has)f(the)g(same)g(pro)s(of)f (for)g(analytic)h(and)g(b)s(ounded)g(p)s(erturbations.)0 3595 y(Its)33 b(pro)s(of)f(is)h(giv)m(en)h(in)e(the)h(next)h(section.) 45 b Fe(2)146 3765 y Fu(W)-8 b(e)26 b(remark)g(that)f(the)g (Golden-Thompson)h(inequalit)m(y)i(w)m(as)e(\014rst)f(pro)m(v)m(en)i(b) m(y)f(Araki)g([Ar2)o(].)42 b(The)0 3885 y(pro)s(of)32 b(describ)s(ed)i(in)f(\(12\))f(is)h(due)g(to)f(Donald)g([Don].)0 4174 y Fs(5.3)135 b(Bounded)44 b(p)t(erturbations|pro)t(ofs)0 4359 y Fu(In)35 b(this)h(subsection)h(w)m(e)f(pro)m(v)m(e)g(Theorem)g (5.1.)50 b(W)-8 b(e)36 b(assume)g(that)f Fo(Q)g Fu(is)g(an)g(arbitrary) g(self-adjoin)m(t)0 4479 y(elemen)m(t)43 b(of)d Fp(M)p Fu(.)69 b(By)41 b(Theorem)i(2.20,)f(w)m(e)g(can)g(\014nd)f(a)g (sequence)j Fo(Q)2637 4494 y Fl(n)2725 4479 y Fu(of)d(self-adjoin)m(t)g Fo(\034)11 b Fu(-analytic)0 4600 y(elemen)m(ts)45 b(suc)m(h)g(that)d Fo(Q)938 4615 y Fl(n)1031 4600 y Fn(!)j Fo(Q)e(\033)t Fu(-strongly)-8 b(.)75 b(This)44 b(implies)g(that)f Fo(Q)2699 4615 y Fl(n)2791 4600 y Fn(!)i Fo(Q)e Fu(strongly)h(and)f(the)0 4720 y(follo)m(wing)33 b(lemma)g(holds:)0 4949 y Fj(Lemma)39 b(5.4)90 b Fu(\(1\))49 b Fo(L)22 b Fu(+)g Fo(Q)1067 4964 y Fl(n)1142 4949 y Fn(!)27 b Fo(L)c Fu(+)f Fo(Q)35 b Fg(in)g(the)f(str)-5 b(ong)35 b(r)-5 b(esolvent)34 b(sense.)41 5110 y Fu(\(2\))48 b Fo(L)280 5125 y Fl(Q)336 5133 y Fb(n)411 5110 y Fn(!)27 b Fo(L)604 5125 y Fl(Q)699 5110 y Fg(in)34 b(the)h(str)-5 b(ong)35 b(r)-5 b(esolvent)34 b(sense.)1841 5753 y Fu(29)p eop %%Page: 30 30 30 29 bop 0 407 a Fj(Pro)s(of)45 b(of)g(Theorem)h(5.1.)64 b Fu(\(1\))39 b(Clearly)-8 b(,)42 b(lim)1839 422 y Fl(n)1902 407 y Fu(e)1945 371 y Fw(\000)p Fl(\014)s(Q)2099 379 y Fb(n)2142 371 y Fl(=)p Fk(2)2216 407 y Fu(\012)e(=)f(e)2484 371 y Fw(\000)p Fl(\014)s(Q=)p Fk(2)2713 407 y Fu(\012.)64 b(Hence)40 b(there)g(exists)i Fo(C)0 527 y Fu(suc)m(h)34 b(that)e(for)g(all)h Fo(n)1532 648 y Fn(k)p Fu(e)1625 606 y Fw(\000)p Fl(\014)s(Q)1779 614 y Fb(n)1821 606 y Fl(=)p Fk(2)1896 648 y Fu(\012)p Fn(k)28 b(\024)g Fo(C)r(:)0 821 y Fu(By)33 b(the)g(Golden-Thompson)h(inequalit)m(y)g(for)e (analytic)h(p)s(erturbations,)1432 1040 y Fn(k)p Fu(\012)1552 1055 y Fl(Q)1608 1063 y Fb(n)1654 1040 y Fn(k)28 b(\024)g(k)p Fu(e)1930 999 y Fw(\000)p Fl(\014)s(Q)2084 1007 y Fb(n)2126 999 y Fl(=)p Fk(2)2201 1040 y Fu(\012)p Fn(k)p Fo(:)0 1259 y Fu(Hence)34 b Fn(k)p Fu(\012)410 1274 y Fl(Q)466 1282 y Fb(n)512 1259 y Fn(k)28 b(\024)g Fo(C)7 b Fu(.)43 b(No)m(w)34 b(b)m(y)f(Prop)s(osition)g(A.6,)f(\012)d Fn(2)f(D)s Fu(\(e)2287 1222 y Fw(\000)p Fl(\014)s Fk(\()p Fl(L)p Fk(+)p Fl(Q)p Fk(\))p Fl(=)p Fk(2)2673 1259 y Fu(\))k(and)1106 1477 y(w)q Fn(\000)43 b Fu(lim)1272 1537 y Fl(n)p Fw(!1)1473 1477 y Fu(e)1516 1436 y Fw(\000)p Fl(\014)s Fk(\()p Fl(L)p Fk(+)p Fl(Q)1800 1444 y Fb(n)1842 1436 y Fk(\))p Fl(=)p Fk(2)1945 1477 y Fu(\012)28 b(=)f(e)2189 1436 y Fw(\000)p Fl(\014)s Fk(\()p Fl(L)p Fk(+)p Fl(Q)p Fk(\))p Fl(=)p Fk(2)2576 1477 y Fu(\012)p Fo(:)146 1730 y Fu(\(2\))32 b(follo)m(ws)i(from)e(the)h(analytic)g(case)h(of)39 b(\(2\))32 b(and)h(the)g(fact)f(that)g Fn(H)2721 1694 y Fk(+)2813 1730 y Fu(is)h(w)m(eakly)i(closed.)146 1850 y(\(3\))d(Let)h Fo(P)41 b Fu(:=)28 b(1)22 b Fn(\000)h Fu(s)922 1865 y Fk(\012)973 1876 y Fb(Q)1029 1850 y Fu(.)44 b(Clearly)-8 b(,)34 b Fo(P)41 b Fn(2)28 b Fp(M)p Fu(,)k Fo(\034)1873 1814 y Fl(t)1862 1875 y(Q)1922 1850 y Fu(\()p Fo(P)14 b Fu(\))27 b(=)g Fo(P)46 b Fu(and)33 b Fo(P)14 b Fu(\012)2651 1865 y Fl(Q)2738 1850 y Fu(=)28 b(0.)43 b(Set)1502 2086 y(\012\()p Fo(z)t Fu(\))28 b(=)g(e)1872 2045 y Fw(\000)p Fl(z)s Fk(\()p Fl(L)p Fk(+)p Fl(Q)p Fk(\))2180 2086 y Fu(\012)p Fo(:)0 2305 y Fu(By)f(Prop)s(osition)f (A.2,)i(the)e(v)m(ector-v)-5 b(alued)28 b(function)e(\012\()p Fo(z)t Fu(\))i(is)e(analytic)h(inside)h(the)e(strip)h(0)h Fo(<)f Fu(Re)p Fo(z)33 b(<)0 2425 y(\014)6 b(=)p Fu(2)32 b(and)g(norm)h(con)m(tin)m(uous)h(on)f(its)g(closure.)45 b(Moreo)m(v)m(er,)34 b(\012\()p Fo(\014)6 b(=)p Fu(2\))27 b(=)g(\012)2719 2440 y Fl(Q)2812 2425 y Fu(and)866 2650 y(e)909 2609 y Fk(i)p Fl(t)p Fk(\()p Fl(L)p Fk(+)p Fl(Q)p Fk(\))1172 2650 y Fo(P)14 b Fu(\012\(i)p Fo(t)22 b Fu(+)g Fo(\014)6 b(=)p Fu(2\))27 b(=)g(e)1909 2609 y Fk(i)p Fl(t)p Fk(\()p Fl(L)p Fk(+)p Fl(Q)p Fk(\))2172 2650 y Fo(P)14 b Fu(e)2292 2609 y Fw(\000)p Fk(i)p Fl(t)p Fk(\()p Fl(L)p Fk(+)p Fl(Q)p Fk(\))2609 2650 y Fu(\012\()p Fo(\014)6 b(=)p Fu(2\))1763 2866 y(=)27 b Fo(\034)1919 2825 y Fl(t)1908 2891 y(Q)1968 2866 y Fu(\()p Fo(P)14 b Fu(\)\012)2191 2881 y Fl(Q)1763 3092 y Fu(=)27 b Fo(P)14 b Fu(\012)2013 3107 y Fl(Q)2100 3092 y Fu(=)28 b(0)p Fo(:)0 3310 y Fu(Th)m(us,)36 b(for)d(all)g(real)h Fo(t)p Fu(,)g Fo(P)14 b Fu(\012\(i)p Fo(t)23 b Fu(+)g Fo(\014)6 b(=)p Fu(2\))29 b(=)g(0.)47 b(This)34 b(implies)i(that)d Fo(P)14 b Fu(\012\()p Fo(z)t Fu(\))30 b(=)f(0)34 b(for)f(all)h Fo(z)k Fu(in)c(the)g(strip)0 3430 y(0)f Fn(\024)g Fu(Re)p Fo(z)38 b Fn(\024)c Fo(\014)6 b(=)p Fu(2.)52 b(In)36 b(particular,)g Fo(P)14 b Fu(\012\(0\))33 b(=)g Fo(P)14 b Fu(\012)33 b(=)g(0.)52 b(Since)37 b(\012)f(is)g(a)g (separating)g(v)m(ector)h(for)e Fp(M)p Fu(,)0 3551 y Fo(P)44 b Fu(=)30 b(0.)49 b(Hence)35 b(s)667 3566 y Fk(\012)718 3577 y Fb(Q)806 3551 y Fu(=)30 b(1)k(and)h(\012)1257 3566 y Fl(Q)1351 3551 y Fu(is)g(a)f(separating)h(v)m(ector)g(for)f Fp(M)p Fu(.)48 b(Since)35 b(\012)2959 3566 y Fl(Q)3054 3551 y Fu(is)g(separating,)g(\(2\))0 3671 y(and)e(Theorem)h(2.9)e (\(3\))g(imply)i(that)e(\012)1471 3686 y Fl(Q)1564 3671 y Fu(is)h(also)f(cyclic.)146 3791 y(\(4\))g(follo)m(ws)i(from)e(the)h (analytic)g(case)h(of)39 b(\(4\))32 b(and)h(Theorem)h(2.19.)146 3912 y(\(5\),)f(\(7\))f(and)g(\(8\))h(follo)m(w)f(from)h(their)g (analytic)g(v)m(ersions)i(and)d(Theorem)i(4.2.)146 4032 y(\(6\))k(Let)g(no)m(w)g Fo(Q)774 4047 y Fk(1)814 4032 y Fo(;)17 b(Q)935 4047 y Fk(2)1012 4032 y Fu(b)s(e)38 b(t)m(w)m(o)g(self-adjoin)m(t)g(elemen)m(ts)i(and)e Fo(Q)2522 4047 y Fk(1)p Fl(;n)2624 4032 y Fu(,)h Fo(Q)2767 4047 y Fk(2)p Fl(;n)2907 4032 y Fu(the)f(sequences)j(of)c(the)0 4152 y(corresp)s(onding)c(analytic)h(appro)m(ximations.)44 b(Then,)34 b(b)m(y)g(the)f(analytic)g(case)g(of)40 b(\(6\))1358 4371 y(\(\012)1466 4386 y Fl(Q)1522 4395 y Ff(1)p Fb(;m)1634 4371 y Fu(\))1672 4386 y Fl(Q)1728 4395 y Ff(2)p Fb(;n)1851 4371 y Fu(=)27 b(\012)2024 4386 y Fl(Q)2080 4395 y Ff(1)p Fb(;m)2188 4386 y Fk(+)p Fl(Q)2299 4395 y Ff(2)p Fb(;n)2395 4371 y Fo(:)0 4590 y Fu(As)33 b Fo(n)28 b Fn(!)f(1)p Fu(,)33 b(\(\012)625 4605 y Fl(Q)681 4614 y Ff(1)p Fb(;m)793 4590 y Fu(\))831 4605 y Fl(Q)887 4614 y Ff(2)p Fb(;n)1010 4590 y Fn(!)27 b Fu(\(\012)1245 4605 y Fl(Q)1301 4614 y Ff(1)p Fb(;m)1413 4590 y Fu(\))1451 4605 y Fl(Q)1507 4614 y Ff(2)1578 4590 y Fu(w)m(eakly)-8 b(,)34 b(\012)1989 4605 y Fl(Q)2045 4614 y Ff(1)p Fb(;m)2153 4605 y Fk(+)p Fl(Q)2264 4614 y Ff(2)p Fb(;n)2387 4590 y Fn(!)27 b Fu(\012)2584 4605 y Fl(Q)2640 4614 y Ff(1)p Fb(;m)2748 4605 y Fk(+)p Fl(Q)2859 4614 y Ff(2)2930 4590 y Fu(w)m(eakly)-8 b(,)34 b(and)f(so)1415 4808 y(\(\012)1523 4823 y Fl(Q)1579 4832 y Ff(1)p Fb(;m)1691 4808 y Fu(\))1729 4823 y Fl(Q)1785 4832 y Ff(2)1851 4808 y Fu(=)27 b(\012)2024 4823 y Fl(Q)2080 4832 y Ff(1)p Fb(;m)2188 4823 y Fk(+)p Fl(Q)2299 4832 y Ff(2)2337 4808 y Fo(:)1166 b Fu(\(5.25\))146 5027 y(By)49 b(the)g(argumen)m(ts)h(of)e(the)h(pro)s(of)f(of)55 b(\(1\),)d(as)d Fo(m)55 b Fn(!)g(1)p Fu(,)d(\012)2590 5042 y Fl(Q)2646 5051 y Ff(1)p Fb(;m)2754 5042 y Fk(+)p Fl(Q)2865 5051 y Ff(2)2958 5027 y Fn(!)i Fu(\012)3182 5042 y Fl(Q)3238 5051 y Ff(1)3273 5042 y Fk(+)p Fl(Q)3384 5051 y Ff(2)3471 5027 y Fu(w)m(eakly)-8 b(.)0 5147 y(Moreo)m(v)m(er,)992 5268 y(\(\012)1100 5283 y Fl(Q)1156 5292 y Ff(1)p Fb(;m)1268 5268 y Fu(\))1306 5283 y Fl(Q)1362 5292 y Ff(2)1428 5268 y Fu(=)28 b(e)1575 5227 y Fw(\000)p Fl(\014)s Fk(\()p Fl(L)p Fk(+)p Fl(Q)1859 5236 y Ff(1)p Fb(;m)1967 5227 y Fw(\000)p Fl(J)6 b(Q)2123 5236 y Ff(1)p Fb(;m)2230 5227 y Fl(J)g Fk(+)p Fl(Q)2386 5236 y Ff(2)2420 5227 y Fk(\))p Fl(=)p Fk(2)2522 5268 y Fu(\012)2592 5283 y Fl(Q)2648 5292 y Ff(1)p Fb(;m)2760 5268 y Fo(;)1841 5753 y Fu(30)p eop %%Page: 31 31 31 30 bop 0 407 a Fu(\012)70 422 y Fl(Q)126 431 y Ff(1)p Fb(;m)266 407 y Fn(!)27 b Fu(\012)463 422 y Fl(Q)519 431 y Ff(1)589 407 y Fu(w)m(eakly)34 b(and)d Fo(L)21 b Fu(+)f Fo(Q)1359 422 y Fk(1)p Fl(;m)1500 407 y Fn(\000)h Fo(J)9 b(Q)1738 422 y Fk(1)p Fl(;m)1860 407 y Fo(J)29 b Fu(+)20 b Fo(Q)2116 422 y Fk(2)2183 407 y Fn(!)27 b Fo(L)21 b Fu(+)f Fo(Q)2570 422 y Fk(1)2629 407 y Fn(\000)h Fo(J)9 b(Q)2867 422 y Fk(1)2907 407 y Fo(J)29 b Fu(+)20 b Fo(Q)3163 422 y Fk(2)3234 407 y Fu(in)31 b(the)h(strong)0 527 y(resolv)m(en)m(t)j(sense.)45 b(Hence)34 b(b)m(y)f(Prop)s(osition)g (A.6,)g(\012)1927 542 y Fl(Q)1983 551 y Ff(1)2049 527 y Fn(2)28 b(D)s Fu(\(e)2304 491 y Fw(\000)p Fl(\014)s Fk(\()p Fl(L)p Fk(+)p Fl(Q)2588 500 y Ff(1)2623 491 y Fw(\000)p Fl(J)6 b(Q)2779 500 y Ff(1)2812 491 y Fl(J)g Fk(+)p Fl(Q)2968 500 y Ff(2)3002 491 y Fk(\))p Fl(=)p Fk(2)3137 527 y Fu(and)918 706 y(\(\012)1026 721 y Fl(Q)1082 730 y Ff(1)1120 706 y Fu(\))1158 721 y Fl(Q)1214 730 y Ff(2)1280 706 y Fu(=)28 b Fo(e)1429 665 y Fw(\000)p Fl(\014)s Fk(\()p Fl(L)p Fk(+)p Fl(Q)1713 674 y Ff(1)1747 665 y Fw(\000)p Fl(J)6 b(Q)1903 674 y Ff(1)1937 665 y Fl(J)g Fk(+)p Fl(Q)2093 674 y Ff(2)2127 665 y Fk(\))p Fl(=)p Fk(2)2229 706 y Fu(\012)2299 721 y Fl(Q)2355 730 y Ff(1)2421 706 y Fu(=)28 b(\012)2595 721 y Fl(Q)2651 730 y Ff(1)2686 721 y Fk(+)p Fl(Q)2797 730 y Ff(2)2835 706 y Fo(:)146 885 y Fu(\(9\))k(and)h(\(10\))f(follo)m(w)h(from)f (\(7\))g(and)h(\(8\))f(precisely)j(as)e(in)g(the)g(analytic)g(case.)146 1005 y(\(11\))f(\(The)i(P)m(eierls-Bogoliub)s(o)m(v)g(inequalit)m(y\))g (follo)m(ws)g(from)e(\(9\))g(just)h(as)g(in)f(the)h(analytic)g(case.) 146 1126 y(\(12\))f(lim)488 1141 y Fl(n)551 1126 y Fu(e)594 1089 y Fw(\000)p Fl(\014)s(Q)748 1097 y Fb(n)791 1089 y Fl(=)p Fk(2)865 1126 y Fu(\012)c(=)g(e)1110 1089 y Fw(\000)p Fl(\014)s(Q=)p Fk(2)1338 1126 y Fu(\012)33 b(implies)1272 1304 y(lim)1248 1364 y Fl(n)p Fw(!1)1448 1304 y Fn(k)p Fu(e)1541 1263 y Fw(\000)p Fl(\014)s(Q)1695 1271 y Fb(n)1737 1263 y Fl(=)p Fk(2)1812 1304 y Fu(\012)p Fn(k)28 b Fu(=)f Fn(k)p Fu(e)2156 1263 y Fw(\000)p Fl(\014)s(Q=)p Fk(2)2385 1304 y Fu(\012)p Fn(k)p Fo(:)998 b Fu(\(5.26\))0 1510 y(Moreo)m(v)m(er,)34 b(w)q Fn(\000)18 b Fu(lim)757 1525 y Fl(n)820 1510 y Fu(\012)890 1525 y Fl(Q)946 1533 y Fb(n)1020 1510 y Fu(=)28 b(\012)1194 1525 y Fl(Q)1287 1510 y Fu(implies)1415 1689 y Fn(k)p Fu(\012)1535 1704 y Fl(Q)1595 1689 y Fn(k)g(\024)g Fu(lim)17 b(inf)1821 1749 y Fl(n)p Fw(!1)2065 1689 y Fn(k)p Fu(\012)2185 1704 y Fl(Q)2241 1712 y Fb(n)2287 1689 y Fn(k)p Fo(:)1166 b Fu(\(5.27\))0 1895 y(By)33 b(the)g(Golden-Thompson)h(inequalit)m(y)g (for)e(analytic)h(p)s(erturbations,)1432 2074 y Fn(k)p Fu(\012)1552 2089 y Fl(Q)1608 2097 y Fb(n)1654 2074 y Fn(k)28 b(\024)g(k)p Fu(e)1930 2032 y Fw(\000)p Fl(\014)s(Q)2084 2040 y Fb(n)2126 2032 y Fl(=)p Fk(2)2201 2074 y Fu(\012)p Fn(k)p Fo(:)1182 b Fu(\(5.28\))0 2252 y(No)m(w)33 b(\(5.26\),)f (\(5.27\))g(and)h(\(5.28\))e(imply)j(the)f(Golden-Thompson)h(inequalit) m(y:)1474 2431 y Fn(k)p Fu(\012)1594 2446 y Fl(Q)1654 2431 y Fn(k)28 b(\024)g(k)p Fu(e)1930 2390 y Fw(\000)p Fl(\014)s(Q=)p Fk(2)2158 2431 y Fu(\012)p Fn(k)p Fo(:)1225 b Fu(\(5.29\))146 2610 y(\(13\))38 b(Let)h Fo(Q)616 2625 y Fl(n)701 2610 y Fn(2)g Fp(M)f Fu(b)s(e)h(an)f(arbitrary)h(sequence)i (of)d(self-adjoin)m(t)h(elemen)m(ts)i(whic)m(h)f(con)m(v)m(erges)0 2730 y(strongly)34 b(to)f Fo(Q)p Fu(.)47 b(The)34 b(pro)s(of)f(of)g (\(1\))g(yields)i(that)e(\012)1933 2745 y Fl(Q)1989 2753 y Fb(n)2065 2730 y Fn(!)c Fu(\012)2264 2745 y Fl(Q)2358 2730 y Fu(w)m(eakly)-8 b(.)48 b(Using)34 b(\014rst)g(the)g(c)m(hain)g (rule)0 2851 y(and)f(then)g(the)g(Golden-Thompson)g(inequalit)m(y)i(w)m (e)f(get)995 3030 y Fn(k)p Fu(\012)1115 3045 y Fl(Q)1171 3053 y Fb(n)1217 3030 y Fn(k)27 b Fu(=)h Fn(k)p Fu(\(\012)1556 3045 y Fl(Q)1616 3030 y Fu(\))1654 3045 y Fl(Q)1710 3053 y Fb(n)1752 3045 y Fw(\000)p Fl(Q)1866 3030 y Fn(k)g(\024)g(k)p Fu(e)2142 2989 y Fw(\000)p Fl(\014)s Fk(\()p Fl(Q)2323 2997 y Fb(n)2365 2989 y Fw(\000)p Fl(Q)p Fk(\))p Fl(=)p Fk(2)2578 3030 y Fu(\012)2648 3045 y Fl(Q)2708 3030 y Fn(k)p Fo(:)0 3208 y Fu(Hence,)42 b(lim)17 b(sup)624 3232 y Fl(n)688 3208 y Fn(k)p Fu(\012)808 3223 y Fl(Q)864 3231 y Fb(n)910 3208 y Fn(k)39 b(\024)g(k)p Fu(\012)1235 3223 y Fl(Q)1295 3208 y Fn(k)p Fu(.)63 b(Com)m(bining)41 b(this)f(estimate)g(with)g(\(5.27\))e(w)m(e)j(get)e Fn(k)p Fu(\012)3489 3223 y Fl(Q)3545 3231 y Fb(n)3591 3208 y Fn(k)g(!)0 3329 y(k)p Fu(\012)120 3344 y Fl(Q)180 3329 y Fn(k)p Fu(,)30 b(and)h(so)f(\012)662 3344 y Fl(Q)718 3352 y Fb(n)792 3329 y Fn(!)e Fu(\012)990 3344 y Fl(Q)1080 3329 y Fu(in)i(norm.)43 b(By)30 b(Theorem)i(2.9,)e(this)h(implies)h (that)e Fo(!)3012 3344 y Fl(Q)3068 3352 y Fb(n)3141 3329 y Fn(!)d Fo(!)3329 3344 y Fl(Q)3419 3329 y Fu(in)j(norm.)0 3449 y Fe(2)0 3770 y Fs(5.4)135 b(P)l(erturbativ)l(e)47 b(expansion)e(of)h FC(\012)1959 3788 y FA(Q)2073 3770 y Fs(and)e(the)i(estimates)0 3955 y Fu(In)32 b(this)h(subsection)g(w)m (e)g(pro)m(v)m(e)g(Theorems)h(5.2)d(and)h(5.3.)43 b(The)33 b(pro)s(of)e(of)g(Theorem)i(5.2)f(is)g(based)h(on)0 4075 y(the)g(follo)m(wing)g(tec)m(hnical)h(result)g(of)e(Araki.)0 4256 y Fj(Theorem)75 b(5.5)90 b Fu(\(1\))49 b Fg(Set)437 4435 y Fo(S)497 4450 y Fl(\014)s(;n)634 4435 y Fu(:=)28 b Fn(f)p Fu(\()p Fo(z)898 4450 y Fk(1)938 4435 y Fo(;)17 b(:)g(:)g(:)e(;)i(z)1201 4450 y Fl(n)1248 4435 y Fu(\))28 b(:)g(Im)p Fo(z)1530 4450 y Fl(i)1586 4435 y Fn(\025)h Fu(0)p Fo(;)86 b(i)28 b Fu(=)f(1)p Fo(;)17 b(:)g(:)g(:)f(;)h(n;)51 b Fu(Im)q Fo(z)2584 4450 y Fk(1)2646 4435 y Fu(+)22 b Fn(\001)17 b(\001)g(\001)j Fu(+)i(Im)q Fo(z)3142 4450 y Fl(n)3217 4435 y Fn(\024)28 b Fo(\014)6 b(=)p Fu(2)p Fn(g)p Fo(:)214 4614 y Fg(Then)35 b(for)f Fu(\()p Fo(z)707 4629 y Fk(1)747 4614 y Fo(;)17 b(:)g(:)g(:)f(;)h(z)1011 4629 y Fl(n)1058 4614 y Fu(\))27 b Fn(2)h Fo(S)1277 4629 y Fl(\014)s(;n)1387 4614 y Fg(,)35 b Fu(\012)g Fg(b)-5 b(elongs)34 b(to)h Fn(D)s Fu(\(e)2169 4578 y Fk(i)p Fl(z)2222 4586 y Fb(n)2264 4578 y Fl(L)2316 4614 y Fo(Q)2393 4629 y Fl(n)2457 4614 y Fn(\001)17 b(\001)g(\001)d Fu(e)2632 4578 y Fk(i)p Fl(z)2685 4587 y Ff(1)2720 4578 y Fl(L)2772 4614 y Fo(Q)2849 4629 y Fk(1)2889 4614 y Fu(\))p Fg(,)34 b(the)h(function)1121 4793 y 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5151 y Fu(\012)p Fn(k)28 b(\024)g(k)p Fo(Q)2591 5166 y Fl(n)2638 5151 y Fn(k)17 b(\001)g(\001)g(\001)d(k)p Fo(Q)2964 5166 y Fk(1)3004 5151 y Fn(k)p Fo(:)449 b Fu(\(5.31\))1841 5753 y(31)p eop %%Page: 32 32 32 31 bop 41 407 a Fu(\(2\))48 b Fg(L)-5 b(et)36 b Fo(Q)460 422 y Fl(i;m)650 407 y Fn(!)598 467 y Fl(m)p Fw(!1)829 407 y Fo(Q)906 422 y Fl(i)969 407 y Fg(str)-5 b(ongly,)35 b Fo(Q)1441 371 y Fw(\003)1441 431 y Fl(i;m)1631 407 y Fn(!)1579 467 y Fl(m)p Fw(!1)1811 407 y Fo(Q)1888 371 y Fw(\003)1888 431 y Fl(i)1962 407 y Fg(str)-5 b(ongly.)45 b(Then)926 669 y Fu(lim)892 729 y Fl(m)p Fw(!1)1112 669 y Fu(e)1155 628 y Fk(i)p Fl(z)1208 636 y Fb(n)1250 628 y Fl(L)1303 669 y Fo(Q)1380 684 y Fl(n;m)1525 669 y Fn(\001)17 b(\001)g(\001)e Fu(e)1701 628 y Fk(i)p Fl(z)1754 637 y Ff(1)1789 628 y Fl(L)1841 669 y Fo(Q)1918 684 y Fk(1)p Fl(;m)2039 669 y Fu(\012)29 b(=)e(e)2284 628 y Fk(i)p Fl(z)2337 636 y Fb(n)2379 628 y Fl(L)2432 669 y Fo(Q)2509 684 y Fl(n)2572 669 y Fn(\001)17 b(\001)g(\001)e Fu(e)2748 628 y Fk(i)p Fl(z)2801 637 y Ff(1)2836 628 y Fl(L)2888 669 y Fo(Q)2965 684 y Fk(1)3004 669 y Fu(\012)p Fo(;)429 b Fu(\(5.32\))214 919 y Fg(uniformly)35 b(for)g Fu(\()p Fo(z)898 934 y Fk(1)937 919 y Fo(;)17 b(:)g(:)g(:)f(;)h(z)1201 934 y Fl(n)1248 919 y Fu(\))35 b Fg(in)g(c)-5 b(omp)g(act)34 b(subsets)g(of)h Fo(S)2320 934 y Fl(\014)s(;n)2430 919 y Fg(.)0 1145 y Fj(Pro)s(of.)62 b Fu(The)32 b(pro)s(of)e(follo)m(ws)i (b)m(y)g(induction)g(wrt)g Fo(n)p Fu(.)43 b(F)-8 b(or)30 b Fo(n)e Fu(=)g(1,)j(the)h(statemen)m(t)g(follo)m(ws)g(from)f(the)0 1265 y(Prop)s(osition)i(A.2)f(and)h(the)g(KMS)g(condition)g(\(Theorem)h (2.18\).)146 1386 y(Supp)s(ose)g(that)e(the)h(statemen)m(t)h(is)f(true) g(for)f Fo(n)23 b Fn(\000)f Fu(1.)44 b(Set)813 1595 y(\012\()p Fo(z)966 1610 y Fk(1)1006 1595 y Fo(;)17 b(:)g(:)g(:)e(;)i(z)1269 1610 y Fl(n)p Fw(\000)p Fk(1)1407 1595 y Fu(\))27 b(:=)h Fo(Q)1680 1610 y Fl(n)1727 1595 y Fu(e)1770 1559 y Fk(i)p Fl(z)1823 1568 y Fb(n)p Fd(\000)p Ff(1)1944 1559 y Fl(L)1996 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(norm)g(con)m(tin)m(uous)h(on)f(the)g(whole)g Fo(S)3040 1549 y Fl(\014)s(;n)3150 1534 y Fu(.)146 1655 y(Next)h(w)m(e)f(turn)g (to)f(the)h(pro)s(of)f(of)39 b(\(2\))32 b(for)g Fo(n)p Fu(.)44 b(Set)676 1849 y(\012)746 1864 y Fl(m)813 1849 y Fu(\()p Fo(z)896 1864 y Fk(1)936 1849 y Fo(;)17 b(:)g(:)g(:)e(;)i(z) 1199 1864 y Fl(n)p Fw(\000)p Fk(1)1337 1849 y Fu(\))27 b(:=)h Fo(Q)1610 1864 y Fl(n;m)1739 1849 y Fu(e)1782 1813 y Fk(i)p Fl(z)1835 1822 y Fb(n)p Fd(\000)p Ff(1)1956 1813 y Fl(L)2008 1849 y Fo(Q)2085 1864 y Fl(n)p Fw(\000)p Fk(1)p Fl(;m)2321 1849 y Fn(\001)17 b(\001)g(\001)e Fu(e)2497 1813 y Fk(i)p Fl(z)2550 1822 y Ff(1)2584 1813 y Fl(L)2636 1849 y Fo(Q)2713 1864 y Fk(1)p Fl(;m)2835 1849 y Fu(\012)p Fo(;)676 2040 y Fu(\012)746 2004 y Fw(\003)746 2065 y Fl(m)813 2040 y Fu(\()p Fo(z)896 2055 y Fk(1)936 2040 y Fo(;)i(:)g(:)g(:)e(;)i(z)1199 2055 y Fl(n)p Fw(\000)p Fk(1)1337 2040 y Fu(\))27 b(:=)h Fo(J)9 b(Q)1673 2004 y Fw(\003)1673 2065 y Fk(1)p Fl(;m)1795 2040 y Fu(e)1838 2004 y Fw(\000)p Fk(i)p 1913 1966 V Fl(z)1948 2013 y Ff(1)1983 2004 y Fl(L)2035 2040 y Fo(Q)2112 2004 y Fw(\003)2112 2065 y Fk(2)p Fl(;m)2250 2040 y Fn(\001)17 b(\001)g(\001)e Fu(e)2426 2004 y Fw(\000)p Fk(i)p 2501 1966 V Fl(z)2537 2013 y Fb(n)p Fd(\000)p Ff(1)2657 2004 y Fl(L)2710 2040 y Fo(Q)2787 2004 y Fw(\003)2787 2065 y Fl(n)p Fw(\000)p Fk(1)p Fl(;m)3006 2040 y Fu(\012)p Fo(:)146 2232 y Fu(By)33 b(the)g(uniform)g(b)s(oundedness)i(principle,)f(indep)s(enden)m(tly)i (of)c Fo(m)p Fu(,)h(w)m(e)g(ha)m(v)m(e)1331 2429 y Fn(k)p Fo(Q)1458 2444 y Fl(i;m)1568 2429 y Fn(k)28 b(\024)g Fo(C)r(;)81 b(i)28 b Fu(=)g(1)p Fo(;)17 b(:)g(:)g(:)e(;)i(n:)1082 b Fu(\(5.37\))146 2626 y(No)m(w)378 2795 y Fn(k)p Fu(\012)498 2759 y Fw(\003)498 2820 y Fl(m)564 2795 y Fu(\()p Fo(z)647 2810 y Fk(1)687 2795 y Fo(;)17 b(:)g(:)g(:)f(;)h(z)951 2810 y Fl(n)p Fw(\000)p Fk(1)1088 2795 y Fu(\))22 b Fn(\000)h Fu(\012)1318 2759 y Fw(\003)1358 2795 y Fu(\()p Fo(z)1441 2810 y Fk(1)1480 2795 y Fo(;)17 b(:)g(:)g(:)f(;)h(z)1744 2810 y Fl(n)p Fw(\000)p Fk(1)1881 2795 y Fu(\))p Fn(k)378 2986 y(\024)28 b(k)p Fo(Q)610 3001 y Fk(1)p Fl(;m)731 2986 y Fn(kk)p Fu(e)874 2950 y Fw(\000)p Fk(i)p 949 2913 V Fl(z)984 2959 y Ff(1)1019 2950 y Fl(L)1071 2986 y Fo(Q)1148 2950 y Fw(\003)1148 3011 y Fk(2)p Fl(;m)1286 2986 y Fn(\001)17 b(\001)g(\001)e Fu(e)1462 2950 y Fw(\000)p Fk(i)p 1537 2913 V Fl(z)1573 2959 y Fb(n)p Fd(\000)p Ff(1)1694 2950 y Fl(L)1746 2986 y Fo(Q)1823 2950 y Fw(\003)1823 3011 y Fl(n)p Fw(\000)p Fk(1)p Fl(;m)2042 2986 y Fu(\012)23 b Fn(\000)f Fu(e)2277 2950 y Fw(\000)p Fk(i)p 2352 2913 V Fl(z)2388 2959 y Ff(1)2422 2950 y Fl(L)2475 2986 y Fo(Q)2552 2950 y Fw(\003)2552 3011 y Fk(2)2608 2986 y Fn(\001)17 b(\001)g(\001)e Fu(e)2784 2950 y Fw(\000)p Fk(i)p 2859 2913 V Fl(z)2894 2959 y Fb(n)p Fd(\000)p Ff(1)3015 2950 y Fl(L)3067 2986 y Fo(Q)3144 2950 y Fw(\003)3144 3011 y Fl(n)p Fw(\000)p Fk(1)3282 2986 y Fu(\012)p Fn(k)378 3178 y Fu(+)p Fn(k)p Fu(\()p Fo(Q)619 3142 y Fw(\003)619 3202 y Fk(1)p Fl(;m)762 3178 y 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b(con)m(v)m(ergence)i Fo(Q)915 3694 y Fw(\003)915 3755 y Fk(1)p Fl(;m)1064 3731 y Fn(!)28 b Fo(Q)1269 3694 y Fw(\003)1269 3755 y Fk(1)1308 3731 y Fu(.)146 3851 y(By)33 b(the)g(pro)s(of)f(of)g(\(1\))g (\(see)i(the)f(iden)m(tit)m(y)h(\(5.36\)\),)e(w)m(e)i(ha)m(v)m(e)g(for) e Fo(z)2624 3866 y Fk(1)2664 3851 y Fo(;)17 b(:)g(:)g(:)e(;)i(z)2927 3866 y Fl(n)p Fw(\000)p Fk(1)3092 3851 y Fn(2)28 b Fo(S)3246 3866 y Fl(\014)s(;n)p Fw(\000)p Fk(1)3446 3851 y Fu(,)610 4048 y(\012\()p Fo(z)763 4063 y Fk(1)803 4048 y Fo(;)17 b(:)g(:)g(:)f(;)h(z)1067 4063 y Fl(n)p Fw(\000)p Fk(1)1204 4048 y Fu(\))22 b Fn(\000)h Fu(\012)1434 4063 y Fl(m)1501 4048 y Fu(\()p Fo(z)1584 4063 y Fk(1)1624 4048 y Fo(;)17 b(:)g(:)g(:)e(;)i(z)1887 4063 y Fl(n)p Fw(\000)p Fk(1)2025 4048 y Fu(\))27 b Fn(2)h(D)s Fu(\(e)2345 4007 y Fk(\()p Fw(\000)p Fk(i)p Fl(z)2480 4016 y Ff(1)2514 4007 y Fw(\000\001\001\001) o(\000)p Fk(i)p Fl(z)2736 4016 y Fb(n)p Fd(\000)p Ff(1)2856 4007 y Fw(\000)p Fl(\014)s(=)p Fk(2\))p Fl(L)3104 4048 y 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y(\015)744 4827 y Fu(e)787 4786 y Fk(\()p Fw(\000)p Fk(i)p Fl(z)922 4795 y Ff(1)957 4786 y Fw(\000\001\001\001)o(\000)p Fk(i)p Fl(z)1179 4795 y Fb(n)p Fd(\000)p Ff(1)1299 4786 y Fw(\000)p Fl(\014)s(=)p Fk(2\))p Fl(L)1547 4827 y Fu(\(\012\()p Fo(z)1738 4842 y Fk(1)1778 4827 y Fo(;)g(:)g(:)g(:)f(;)h(z)2042 4842 y Fl(n)p Fw(\000)p Fk(1)2179 4827 y Fu(\))22 b Fn(\000)h Fu(\012)2409 4842 y Fl(m)2476 4827 y Fu(\()p Fo(z)2559 4842 y Fk(1)2598 4827 y Fo(;)17 b(:)g(:)g(:)f(;)h(z)2862 4842 y Fl(n)p Fw(\000)p Fk(1)2999 4827 y Fu(\)\))3075 4742 y Fc(\015)3075 4802 y(\015)3158 4827 y Fu(=)28 b(0)0 5051 y(uniformly)33 b(on)g(compact)g(subsets)i(of)d Fo(S)1479 5066 y Fl(\014)s(;n)p Fw(\000)p Fk(1)1679 5051 y Fu(.)43 b(By)33 b(the)g(induction)h(assumption,)948 5248 y(lim)914 5308 y Fl(m)p Fw(!1)1134 5248 y Fn(k)p Fu(\012\()p Fo(z)1337 5263 y Fk(1)1377 5248 y Fo(;)17 b(:)g(:)g(:)f(;)h(z)1641 5263 y Fl(n)p Fw(\000)p Fk(1)1778 5248 y Fu(\))22 b Fn(\000)g Fu(\012)2007 5263 y Fl(m)2074 5248 y Fu(\()p Fo(z)2157 5263 y Fk(1)2197 5248 y Fo(;)17 b(:)g(:)g(:)f(;)h(z)2461 5263 y Fl(n)p Fw(\000)p Fk(1)2598 5248 y Fu(\))p Fn(k)28 b Fu(=)f(0)1841 5753 y(33)p eop %%Page: 34 34 34 33 bop 0 407 a Fu(uniformly)33 b(on)g(compact)g(subsets)i(of)d Fo(S)1479 422 y Fl(\014)s(;n)p Fw(\000)p Fk(1)1679 407 y Fu(.)43 b(Hence,)34 b(b)m(y)f(Prop)s(osition)g(A.2)787 657 y(lim)753 717 y Fl(m)p Fw(!1)974 543 y Fc(\015)974 602 y(\015)974 662 y(\015)1029 657 y Fu(e)1072 616 y Fk(i)p Fl(z)1125 624 y Fb(n)1167 616 y Fl(L)1219 547 y Fc(\020)1279 657 y Fu(\012\()p Fo(z)1432 672 y Fk(1)1472 657 y Fo(;)17 b(:)g(:)g(:)f(;)h(z)1736 672 y Fl(n)p Fw(\000)p Fk(1)1873 657 y Fu(\))22 b Fn(\000)h Fu(\012)2103 672 y Fl(m)2170 657 y Fu(\()p Fo(z)2253 672 y Fk(1)2292 657 y Fo(;)17 b(:)g(:)g(:)f(;)h(z)2556 672 y Fl(n)p Fw(\000)p Fk(1)2693 657 y Fu(\))2731 547 y Fc(\021)2791 543 y(\015)2791 602 y(\015)2791 662 y(\015)2874 657 y Fu(=)27 b(0)0 910 y(uniformly)h(for)f(0)g Fn(\024)h Fu(Im)q Fo(z)926 925 y Fl(n)1001 910 y Fn(\024)g Fo(\014)6 b(=)p Fu(2)11 b Fn(\000)g Fu(Im)q Fo(z)1526 925 y Fk(1)1577 910 y Fn(\000)g(\001)17 b(\001)g(\001)10 b(\000)h Fu(Im)q Fo(z)2043 925 y Fl(n)p Fw(\000)p Fk(1)2208 910 y Fu(and)27 b(\()p Fo(z)2475 925 y Fk(1)2515 910 y Fo(;)17 b(:)g(:)g(:)f(;)h(z)2779 925 y Fl(n)p Fw(\000)p Fk(1)2916 910 y Fu(\))27 b(in)h(compact)f (subsets)0 1030 y(of)g Fo(S)166 1045 y Fl(\014)s(;n)p Fw(\000)p Fk(1)366 1030 y Fu(.)42 b(In)28 b(particular,)h(the)g(con)m (v)m(ergence)h(is)e(uniform)g(on)g(compact)h(subsets)h(of)d Fo(S)3193 1045 y Fl(\014)s(;n)3302 1030 y Fu(.)42 b(This)29 b(ends)0 1150 y(the)k(pro)s(of)f(of)39 b(\(2\))32 b(for)g Fo(n)p Fu(.)146 1271 y(It)23 b(remains)h(to)e(pro)m(v)m(e)i(the)f(norm) f(con)m(tin)m(uit)m(y)j(part)d(of)g(\(1\).)40 b(Let)23 b Fo(Q)2547 1286 y Fl(i;m)2685 1271 y Fn(2)28 b Fp(M)2884 1286 y Fl(\034)2949 1271 y Fu(suc)m(h)c(that)f Fo(Q)3438 1286 y Fl(i;m)3628 1271 y Fn(!)3576 1330 y Fl(m)p Fw(!1)0 1418 y Fo(Q)77 1433 y Fl(i)138 1418 y Fu(strongly)33 b(and)g Fo(Q)782 1382 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y(that)g(\(5.30\))g(is)h(norm)g(con)m (tin)m(uous)h(on)f Fo(S)1530 2158 y Fl(\014)s(;n)1639 2143 y Fu(.)44 b Fe(2)0 2313 y Fj(Pro)s(of)g(of)h(Theorem)h(5.2.)62 b Fu(Let)39 b Fo(Q)1428 2328 y Fl(n)1514 2313 y Fn(2)g Fp(M)1724 2328 y Fl(\034)1806 2313 y Fu(b)s(e)g(suc)m(h)h(that)f Fo(Q)2466 2328 y Fl(n)2551 2313 y Fn(!)f Fo(Q)i Fu(strongly)-8 b(.)63 b(Since)40 b(\012)3563 2328 y Fl(Q)3619 2336 y Fb(n)3704 2313 y Fu(=)0 2433 y(E)66 2448 y Fl(Q)122 2456 y Fb(n)169 2433 y Fu(\(i)p Fo(\014)6 b(=)p Fu(2\)\012,)32 b(the)h(expansion)h(\(3.9\))e(yields)i(that)e(Theorem)i(5.2)e(holds)i (for)e Fo(Q)2950 2448 y Fl(n)2997 2433 y Fu(.)43 b(Moreo)m(v)m(er,)483 2654 y(\012)553 2669 y Fl(Q)641 2654 y Fu(=)27 b(w)q Fn(\000)43 b Fu(lim)910 2713 y Fl(n)p Fw(!1)1111 2654 y Fu(\012)1181 2669 y Fl(Q)1237 2677 y Fb(n)641 2990 y Fu(=)27 b(w)q Fn(\000)43 b Fu(lim)910 3050 y Fl(n)p Fw(!1)1152 2866 y(1)1115 2896 y Fc(X)1111 3105 y Fl(m)p Fk(=0)1263 2990 y Fu(\()p Fn(\000)p Fu(1\))1465 2949 y Fl(m)1532 2855 y Fc(Z)1632 2990 y Fn(\001)17 b(\001)g(\001)1748 2855 y Fc(Z)1612 3138 y Fl(T)1653 3150 y Fb(\014)s(;m)1848 2990 y Fu(e)1891 2949 y Fw(\000)p Fl(\014)1986 2958 y Ff(1)2020 2949 y Fl(L)2072 2990 y Fo(Q)2149 3005 y Fl(n)2213 2990 y Fn(\001)g(\001)g(\001)e Fu(e)2389 2949 y Fw(\000)p Fl(\014)2484 2957 y Fb(m)2542 2949 y Fl(L)2595 2990 y Fo(Q)2672 3005 y Fl(n)2735 2990 y Fu(\012)i(d)p Fo(\014)2931 3005 y Fk(1)2988 2990 y Fn(\001)g(\001)g(\001)e Fu(d)p Fo(\014)3230 3005 y Fl(m)641 3437 y Fu(=)785 3312 y Fw(1)749 3342 y Fc(X)744 3551 y Fl(m)p Fk(=0)897 3437 y Fu(\()p Fn(\000)p Fu(1\))1099 3396 y Fl(m)1166 3301 y Fc(Z)1265 3437 y Fn(\001)i(\001)g(\001)1382 3301 y Fc(Z)1246 3585 y Fl(T)1287 3597 y Fb(\014)s(;m)1481 3437 y Fu(e)1524 3396 y Fw(\000)p Fl(\014)1619 3405 y Ff(1)1654 3396 y Fl(L)1706 3437 y Fo(Q)g Fn(\001)g(\001)g(\001)d Fu(e)1975 3396 y Fw(\000)p Fl(\014)2070 3404 y Fb(m)2129 3396 y Fl(L)2181 3437 y Fo(Q)j Fu(\012)g(d)p Fo(\014)2471 3452 y Fk(1)2527 3437 y Fn(\001)g(\001)g(\001)e Fu(d)p Fo(\014)2769 3452 y Fl(m)2836 3437 y Fo(:)0 3799 y Fu(The)27 b(\014rst)g(iden)m(tit) m(y)h(follo)m(ws)f(from)f(Theorem)h(5.1)f(\(recall)h(the)g(pro)s(of)e (of)h(\(1\))g(or)f(use)j(\(13\)\),)f(the)f(second)0 3919 y(is)33 b(ob)m(vious,)h(and)f(the)g(third)g(follo)m(ws)g(from)f (Theorem)i(5.5.)43 b Fe(2)0 4089 y Fj(Pro)s(of)48 b(of)g(Theorem)g (5.3.)71 b Fu(Theorem)43 b(5.2)e(yields)j(\(1\).)70 b(By)42 b(Theorem)h(5.1)e(\(13\))g(it)h(su\016ces)i(to)0 4209 y(pro)m(v)m(e)34 b(\(2\){\(5\))d(for)h Fo(Q)c Fn(2)g Fp(M)1046 4224 y Fl(\034)1089 4209 y Fu(.)146 4330 y(\(2\)-\(3\).)49 b(Our)34 b(pro)s(of)g(is)h(motiv)-5 b(ated)35 b(b)m(y)h([Sa2].)50 b(By)35 b(Theorem)h(3.2,)e(\012)e Fn(2)g(D)s Fu(\(e)3061 4294 y Fw(\000)p Fl(z)s Fk(\()p Fl(L)p Fk(+)p Fl(Q)p Fk(\))3369 4330 y Fu(\))i(for)g(all)h Fo(z)0 4450 y Fu(and)1439 4571 y(E)1505 4586 y Fl(Q)1565 4571 y Fu(\(i)p Fo(z)t Fu(\)\012)29 b(=)e(e)1962 4529 y Fw(\000)p Fl(z)s Fk(\()p Fl(L)p Fk(+)p Fl(Q)p Fk(\))2270 4571 y Fu(\012)0 4745 y(is)33 b(an)g(en)m(tire)g(v)m(ector-v)-5 b(alued)34 b(function.)44 b(Set)1080 4965 y Fo(f)11 b Fu(\()p Fo(z)t Fu(\))28 b(:=)f(\(\012)p Fn(j)p Fu(e)1601 4924 y Fw(\000)p Fl(z)s Fk(\()p Fl(L)p Fk(+)p Fl(Q)p Fk(\))1910 4965 y Fu(\012\))h(=)f(\(\012)p Fn(j)p Fu(E)2351 4980 y Fl(Q)2412 4965 y Fu(\(i)p Fo(z)t Fu(\)\012\))p Fo(:)1841 5753 y Fu(34)p eop %%Page: 35 35 35 34 bop 0 407 a Fu(Then)34 b Fo(f)43 b Fu(is)33 b(an)g(en)m(tire)g (function,)g Fo(f)1321 371 y Fw(00)1364 407 y Fu(\()p Fo(x)p Fu(\))28 b Fn(\025)g Fu(0)k(for)g Fo(x)c Fn(2)g Fm(R)p Fu(,)33 b(and)612 624 y Fo(f)11 b Fu(\(0\))27 b(=)g Fn(k)p Fu(\012)p Fn(k)1096 588 y Fk(2)1163 624 y Fu(=)h(1)p Fo(;)114 b(f)11 b Fu(\()p Fo(\014)6 b(=)p Fu(2\))26 b(=)i(\(\012)p Fn(j)p Fu(\012)2087 639 y Fl(Q)2147 624 y Fu(\))p Fo(;)114 b(f)11 b Fu(\()p Fo(\014)6 b Fu(\))27 b(=)h Fn(k)p Fu(\012)2773 639 y Fl(Q)2833 624 y Fn(k)2883 588 y Fk(2)2922 624 y Fo(;)612 815 y(f)671 779 y Fw(0)694 815 y Fu(\(0\))f(=)g Fn(\000)p Fu(\(\012)p Fn(j)p Fu(\()p Fo(L)c Fu(+)f Fo(Q)p Fu(\)\012\))29 b(=)e Fn(\000)p Fu(\(\012)p Fn(j)p Fo(Q)p Fu(\012\))p Fo(;)612 1006 y(f)671 970 y Fw(0)694 1006 y Fu(\()p Fo(\014)6 b(=)p Fu(2\))26 b(=)i Fn(\000)p Fu(\(\012)p Fn(j)p Fu(\()p Fo(L)23 b Fu(+)f Fo(Q)p Fu(\)\012)1682 1021 y Fl(Q)1742 1006 y Fu(\))28 b(=)f Fn(\000)p Fu(\(\012)p Fn(j)p Fo(J)9 b(QJ)g Fu(\012)2397 1021 y Fl(Q)2459 1006 y Fu(\))27 b(=)h Fn(\000)p Fu(\(\012)p Fn(j)p Fo(Q)p Fu(\012)2988 1021 y Fl(Q)3049 1006 y Fu(\))p Fo(;)612 1198 y(f)671 1161 y Fw(0)694 1198 y Fu(\()p Fo(\014)6 b Fu(\))27 b(=)g Fn(\000)p Fu(\(\012)1146 1213 y Fl(Q)1207 1198 y Fn(j)p Fu(\()p Fo(L)22 b Fu(+)g Fo(Q)p Fu(\)\012)1644 1213 y Fl(Q)1704 1198 y Fu(\))28 b(=)f Fn(\000)p Fu(\(\012)2058 1213 y Fl(Q)2119 1198 y Fn(j)p Fo(J)9 b(QJ)g Fu(\012)2420 1213 y Fl(Q)2480 1198 y Fu(\))28 b(=)f Fn(\000)p Fu(\(\012)2834 1213 y Fl(Q)2895 1198 y Fn(j)p Fo(Q)p Fu(\012)3070 1213 y Fl(Q)3130 1198 y Fu(\))0 1416 y(\(w)m(e)47 b(used)h Fo(L)p Fu(\012)j(=)g(0)46 b(and)g(\()p Fo(L)32 b Fu(+)f Fo(Q)h Fn(\000)g Fo(J)9 b(QJ)g Fu(\)\012)1816 1431 y Fl(Q)1928 1416 y Fu(=)50 b(0\).)84 b(These)48 b(relations)f(com)m(bined)h(with)f(the)0 1537 y(mean-v)-5 b(alue)33 b(theorem)g(yield)h(\(2\){\(3\).)146 1657 y(\(4\))e(follo)m(ws)i(easily)g(from)e(\(2\).)146 1778 y(T)-8 b(o)33 b(pro)m(v)m(e)h(\(5\),)e(consider)i(the)f(function) 1407 1998 y Fo(F)14 b Fu(\()p Fo(z)t Fu(\))28 b(:=)g Fo(\034)1821 1956 y Fl(z)1810 2022 y(Q)1870 1998 y Fu(\()p Fo(Q)p Fu(\)E)2089 2013 y Fl(Q)2149 1998 y Fu(\()p Fo(z)t Fu(\)\012)p Fo(:)0 2218 y Fu(Since)35 b Fo(\034)309 2181 y Fl(z)298 2243 y(Q)358 2218 y Fu(\()p Fo(Q)p Fu(\))g(and)f(E)803 2233 y Fl(Q)863 2218 y Fu(\()p Fo(z)t Fu(\))h(are)f(uniformly)h(b)s (ounded)g(on)f(the)g(strip)h(0)30 b Fn(\024)h Fu(Im)q Fo(z)k Fn(\024)c Fo(\014)6 b(=)p Fu(2,)33 b Fo(F)14 b Fu(\()p Fo(z)t Fu(\))35 b(is)f(also)0 2338 y(b)s(ounded)f(on)g(the)g (this)g(strip.)44 b(Moreo)m(v)m(er,)967 2669 y Fn(k)p Fo(F)14 b Fu(\()p Fo(z)t Fu(\))p Fn(k)28 b(\024)1402 2465 y Fc(8)1402 2555 y(<)1402 2734 y(:)1532 2563 y Fn(k)p Fo(Q)p Fu(\012)p Fn(k)410 b Fu(if)32 b(Im)q Fo(z)g Fu(=)c(0)o(;)1532 2771 y Fn(k)p Fo(\034)1635 2720 y Fk(i)p Fl(\014)s(=)p Fk(2)1624 2798 y Fl(Q)1773 2771 y Fu(\()p Fo(Q)p Fu(\)\012)1996 2786 y Fl(Q)2056 2771 y Fn(k)83 b Fu(if)32 b(Im)q Fo(z)g Fu(=)c Fo(\014)6 b(=)p Fu(2)o Fo(:)0 3033 y Fu(Since)34 b Fo(\034)308 2982 y Fk(i)p Fl(\014)s(=)p Fk(2)297 3060 y Fl(Q)446 3033 y Fu(\()p Fo(Q)p Fu(\)\012)669 3048 y Fl(Q)756 3033 y Fu(=)28 b(e)903 2996 y Fw(\000)p Fl(\014)s(L)1049 3007 y Fb(Q)1102 2996 y Fl(=)p Fk(2)1176 3033 y Fo(Q)p Fu(\012)1323 3048 y Fl(Q)1411 3033 y Fu(=)g Fo(J)9 b(Q)p Fu(\012)1725 3048 y Fl(Q)1785 3033 y Fu(,)1091 3253 y Fn(k)p Fo(F)14 b Fu(\()p Fo(z)t Fu(\))p Fn(k)28 b(\024)g(k)p Fo(Q)p Fu(\012)1723 3268 y Fl(Q)1783 3253 y Fn(k)195 b Fu(if)121 b(Im)p Fo(z)33 b Fu(=)27 b Fo(\014)6 b(=)p Fu(2)p Fo(:)0 3473 y Fu(Hence,)34 b(b)m(y)f(the)g(three-line)h (theorem,)g(for)e(0)27 b Fn(\024)h Fo(t)g Fn(\024)g Fo(\014)6 b(=)p Fu(2,)1187 3693 y Fn(k)p Fo(F)14 b Fu(\(i)p Fo(t)p Fu(\))p Fn(k)27 b(\024)h(k)p Fo(Q)p Fu(\012)1831 3708 y Fl(Q)1892 3693 y Fn(k)1942 3651 y Fk(1)p Fw(\000)p Fk(2)p Fl(t=\014)2175 3693 y Fn(k)p Fo(Q)p Fu(\012)p Fn(k)2422 3651 y Fk(2)p Fl(t=\014)2565 3693 y Fo(:)0 3913 y Fu(Since)1145 4080 y(\012)1215 4095 y Fl(Q)1298 4080 y Fn(\000)22 b Fu(\012)28 b(=)g Fn(\000)1693 3944 y Fc(Z)1793 3971 y Fl(\014)s(=)p Fk(2)1748 4170 y(0)1927 4080 y Fo(\034)1980 4039 y Fk(i)p Fl(t)1969 4105 y(Q)2030 4080 y Fu(\()p Fo(Q)p Fu(\)E)2249 4095 y Fl(Q)2309 4080 y Fu(\(i)p Fo(t)p Fu(\)\012d)p Fo(t;)0 4301 y Fu(w)m(e)34 b(deriv)m(e)550 4513 y Fn(k)p Fu(\012)670 4528 y Fl(Q)752 4513 y Fn(\000)23 b Fu(\012)p Fn(k)83 b(\024)1160 4433 y Fc(R)1227 4459 y Fl(\014)s(=)p Fk(2)1207 4548 y(0)1361 4513 y Fn(k)p Fo(F)14 b Fu(\(i)p Fo(t)p Fu(\))p Fn(k)p Fu(d)p Fo(t)1055 4717 y Fn(\024)1170 4673 y Fl(\014)p 1170 4694 44 4 v 1174 4751 a Fk(2)1240 4637 y Fc(R)1306 4663 y Fk(1)1287 4751 y(0)1362 4717 y Fn(k)p Fo(Q)p Fu(\012)p Fn(k)1609 4681 y Fk(1)p Fw(\000)p Fl(s)1736 4717 y Fn(k)p Fo(Q)p Fu(\012)1933 4732 y Fl(Q)1994 4717 y Fn(k)2044 4681 y Fl(s)2080 4717 y Fu(d)p Fo(s)28 b Fu(=)g Fo(\014)6 b(f)11 b Fu(\()p Fn(k)p Fo(Q)p Fu(\012)p Fn(k)p Fo(;)17 b Fn(k)p Fo(Q)p Fu(\012)2958 4732 y Fl(Q)3017 4717 y Fn(k)p Fu(\))p Fo(=)p Fu(2)p Fo(:)0 4905 y Fe(2)1841 5753 y Fu(35)p eop %%Page: 36 36 36 35 bop 0 407 a Fs(5.5)135 b(Un)l(b)t(ounded)44 b(p)t(erturbations)0 592 y Fu(This)35 b(subsection)h(con)m(tains)f(our)f(main)g(results.)49 b(It)35 b(extends)g(the)g(construction)g(of)f(KMS)g(states)h(to)0 712 y(a)d(large)h(class)g(of)g(un)m(b)s(ounded)h(p)s(erturbations.)146 832 y(Let)h Fo(Q)g Fu(b)s(e)g(a)f(self-adjoin)m(t)h(op)s(erator)f (a\016liated)h(to)f Fp(M)g Fu(satisfying)i(Assumptions)g(3.A)f(and)f (3.B.)0 953 y(Let)43 b Fo(\034)227 968 y Fl(Q)329 953 y Fu(b)s(e)g(the)g(dynamics)i(de\014ned)f(as)f(in)g(Subsection)h(3.3.) 74 b(Recall)43 b(that)f(b)m(y)i(Theorem)g(3.5)e(its)0 1073 y(Liouvillean)34 b(equals)1458 1193 y Fo(L)1524 1208 y Fl(Q)1612 1193 y Fu(=)28 b Fo(L)22 b Fu(+)g Fo(Q)h Fn(\000)f Fo(J)9 b(QJ)n(:)146 1362 y Fu(In)28 b(order)g(to)f(construct) h(the)g(p)s(erturb)s(ed)g(KMS)g(state)f(w)m(e)i(will)f(need)g(an)g (additional)f(assumption:)0 1574 y Fj(Assumption)38 b(5.A)154 b Fn(k)p Fu(e)1040 1538 y Fw(\000)p Fl(\014)s(Q=)p Fk(2)1268 1574 y Fu(\012)p Fn(k)28 b Fo(<)f Fn(1)p Fg(.)0 1787 y Fj(Theorem)75 b(5.6)49 b Fg(Assume)35 b(3.A,)g(3.B)f(and)h(5.A.)44 b(Then)41 1938 y Fu(\(1\))k(\012)29 b Fn(2)f(D)s Fu(\(e)568 1902 y Fw(\000)p Fl(\014)s Fk(\()p Fl(L)p Fk(+)p Fl(Q)p Fk(\))p Fl(=)p Fk(2)953 1938 y Fu(\))p Fg(.)45 b(Set)907 2145 y Fu(\012)977 2160 y Fl(Q)1064 2145 y Fu(:=)28 b(e)1238 2104 y Fw(\000)p Fl(\014)s Fk(\()p Fl(L)p Fk(+)p Fl(Q)p Fk(\))p Fl(=)p Fk(2)1624 2145 y Fu(\012)p Fo(;)157 b(!)1939 2160 y Fl(Q)1998 2145 y Fu(\()p Fo(A)p Fu(\))28 b(:=)g(\(\012)2414 2160 y Fl(Q)2474 2145 y Fn(j)p Fo(A)p Fu(\012)2645 2160 y Fl(Q)2704 2145 y Fu(\))p Fo(=)p Fn(k)p Fu(\012)2911 2160 y Fl(Q)2971 2145 y Fn(k)3021 2104 y Fk(2)3060 2145 y Fo(:)41 2390 y Fu(\(2\))48 b(\012)284 2405 y Fl(Q)372 2390 y Fn(2)28 b(H)551 2354 y Fk(+)610 2390 y Fg(.)41 2550 y Fu(\(3\))48 b(\012)284 2565 y Fl(Q)379 2550 y Fg(is)35 b(cyclic)g(and)f(sep)-5 b(ar)g(ating.)41 2710 y Fu(\(4\))48 b Fo(!)275 2725 y Fl(Q)370 2710 y Fg(is)34 b(a)h Fu(\()p Fo(\034)639 2725 y Fl(Q)699 2710 y Fo(;)17 b(\014)6 b Fu(\))p Fg(-KMS)34 b(state.)41 2870 y Fu(\(5\))48 b(log)18 b(\001)438 2885 y Fk(\012)489 2896 y Fb(Q)573 2870 y Fu(=)28 b Fn(\000)p Fo(\014)6 b(L)881 2885 y Fl(Q)941 2870 y Fg(.)41 3030 y Fu(\(6\))48 b(log)18 b(\001)438 3045 y Fk(\012)489 3056 y Fb(Q)542 3045 y Fl(;)p Fk(\012)644 3030 y Fu(=)28 b Fn(\000)p Fo(\014)6 b(L)22 b Fn(\000)h Fo(\014)6 b(Q)p Fg(.)41 3190 y Fu(\(7\))48 b(En)m(t)q(\()p Fo(!)t Fn(j)p Fo(!)562 3205 y Fl(Q)621 3190 y Fu(\))27 b(=)h Fn(\000)p Fo(\014)6 b(!)t Fu(\()p Fo(Q)p Fu(\))21 b Fn(\000)i Fu(log)18 b Fn(k)p Fu(\012)1530 3205 y Fl(Q)1590 3190 y Fn(k)1640 3154 y Fk(2)1679 3190 y Fg(.)41 3350 y Fu(\(8\))48 b Fg(The)35 b(Peierls-Bo)-5 b(goliub)g(ov)33 b(ine)-5 b(quality)34 b(holds)1578 3556 y Fu(e)1621 3515 y Fw(\000)p Fl(\014)s Fk(\(\012)p Fw(j)p Fl(Q)p Fk(\012\))p Fl(=)p Fk(2)2054 3556 y Fn(\024)28 b(k)p Fu(\012)2279 3571 y Fl(Q)2339 3556 y Fn(k)p Fo(:)41 3802 y Fu(\(9\))48 b Fg(The)35 b(Golden-Thompson)d(ine)-5 b(quality)35 b(holds:)1581 4008 y Fn(k)p Fu(\012)1701 4023 y Fl(Q)1761 4008 y Fn(k)28 b(\024)g(k)p Fu(e)2037 3967 y Fw(\000)p Fl(\014)s(Q=)p Fk(2)2265 4008 y Fu(\012)p Fn(k)p Fo(:)-6 4254 y Fu(\(10\))48 b Fg(F)-7 b(or)36 b(any)h Fu(0)31 b Fn(\024)h Fo(\025)f Fn(\024)h Fu(1)p Fg(,)37 b Fo(\025Q)g Fg(satis\014es)f(the)h (assumptions)f(of)h(the)g(the)-5 b(or)g(em,)36 b(henc)-5 b(e)36 b Fu(\012)3370 4269 y Fl(\025Q)3509 4254 y Fg(is)g(wel)5 b(l)214 4374 y(de\014ne)-5 b(d.)44 b(Mor)-5 b(e)g(over,)35 b Fu(lim)1177 4389 y Fl(\025)p Fw(#)p Fk(0)1310 4374 y Fn(k)p Fu(\012)1430 4389 y Fl(\025Q)1553 4374 y Fn(\000)22 b Fu(\012)p Fn(k)28 b Fu(=)g(0)p Fg(.)0 4618 y Fj(Remark.)62 b Fu(The)39 b(form)m(ula)f(for)g(relativ)m(e)i(en)m(trop)m(y)g(of)45 b(\(7\))38 b(requires)i(a)e(commen)m(t.)63 b(Because)39 b(of)f(As-)0 4738 y(sumption)47 b(5.A,)h Fo(!)t Fu(\()p Fo(Q)846 4753 y Fw(\000)905 4738 y Fu(\))d(is)h(\014nite,)k(where)c Fo(Q)1763 4753 y Fw(\000)1872 4738 y Fu(=)k(1)2047 4754 y Fk(])p Fw(\0001)p Fl(;)p Fk(0])2270 4738 y Fu(\()p Fo(Q)p Fu(\))p Fo(Q)p Fu(.)83 b(Therefore,)50 b Fo(!)t Fu(\()p Fo(Q)p Fu(\))45 b(is)h(a)f(\014nite)0 4859 y(n)m(um)m(b)s(er)34 b(or)e(+)p Fn(1)p Fu(.)146 4979 y(Set)1494 5099 y Fo(Q)1571 5114 y Fl(n)1646 5099 y Fu(:=)27 b(1)1825 5115 y Fk([)p Fw(\000)p Fl(n;n)p Fk(])2028 5099 y Fu(\()p Fo(Q)p Fu(\))p Fo(Q;)0 5268 y Fu(where)34 b(1)331 5283 y Fk([)p Fw(\000)p Fl(n;n)p Fk(])534 5268 y Fu(\()p Fo(Q)p Fu(\))f(is)g(the)g(sp)s(ectral) g(pro)5 b(jection)34 b(of)e Fo(Q)g Fu(on)h(the)g(in)m(terv)-5 b(al)33 b([)p Fn(\000)p Fo(n;)17 b(n)p Fu(].)1841 5753 y(36)p eop %%Page: 37 37 37 36 bop 0 407 a Fj(Lemma)39 b(5.7)90 b Fu(\(1\))49 b Fo(L)22 b Fu(+)g Fo(Q)1067 422 y Fl(n)1142 407 y Fn(!)27 b Fo(L)c Fu(+)f Fo(Q)35 b Fg(in)g(the)f(str)-5 b(ong)35 b(r)-5 b(esolvent)34 b(sense.)41 568 y Fu(\(2\))48 b Fo(L)280 583 y Fl(Q)336 591 y Fb(n)411 568 y Fn(!)27 b Fo(L)604 583 y Fl(Q)699 568 y Fg(in)34 b(the)h(str)-5 b(ong)35 b(r)-5 b(esolvent)34 b(sense.)0 786 y Fj(Pro)s(of.)55 b Fu(W)-8 b(e)29 b(pro)m(v)m(e)g(only)f(\(2\))f(\(the)h(pro)s(of)f(of)h (\(1\))f(is)h(similar\).)43 b(Let)28 b Fn(D)2586 801 y Fk(0)2653 786 y Fu(=)f Fn(D)s Fu(\()p Fo(L)p Fu(\))12 b Fn(\\)g(D)s Fu(\()p Fo(Q)p Fu(\))g Fn(\\)g(D)s Fu(\()p Fo(J)d(QJ)g Fu(\).)0 907 y(By)49 b(Assumption)h(3.B,)i Fo(L)1015 922 y Fl(Q)1123 907 y Fu(is)d(essen)m(tially)i(self-adjoin)m (t)e(on)f Fn(D)2466 922 y Fk(0)2505 907 y Fu(.)91 b(Moreo)m(v)m(er,)54 b Fo(L)3164 922 y Fl(Q)3220 930 y Fb(n)3266 907 y Fu(\011)g Fn(!)h Fo(L)3617 922 y Fl(Q)3677 907 y Fu(\011,)0 1027 y(\011)28 b Fn(2)g(D)275 1042 y Fk(0)314 1027 y Fu(.)43 b(Hence)34 b(the)f(statemen)m(t)h(follo)m(ws)g(from)e(Prop)s(osition)h (A.5.)43 b Fe(2)0 1195 y Fj(Pro)s(of)56 b(of)g(Theorem)h(5.6.)92 b Fu(Giv)m(en)50 b(the)f(appro)m(ximating)h(sequence)h Fo(Q)2881 1210 y Fl(n)2977 1195 y Fu(de\014ned)g(ab)s(o)m(v)m(e)e(and)0 1315 y(Lemma)38 b(2,)h(the)g(parts)f(\(1\)-\(9\))e(follo)m(w)i(from)g (Theorem)h(5.1)e(in)h(the)h(same)f(w)m(a)m(y)h(as)f(the)h(analogous)0 1436 y(parts)33 b(of)f(Theorem)i(5.1)e(follo)m(w)m(ed)i(from)e(the)h (analytic)g(case)h(of)e(Theorem)i(5.1.)146 1556 y(The)25 b(only)f(part)g(requiring)g(a)g(separate)g(argumen)m(t)g(is)h(\(10\).) 39 b(T)-8 b(o)24 b(pro)m(v)m(e)h(it,)h(note)d(that)h Fo(L)t Fu(+)t Fo(\025Q)k Fn(!)f Fo(L)0 1677 y Fu(in)33 b(the)g(strong)f(resolv)m(en)m(t)j(sense)g(as)d Fo(\025)c Fn(#)f Fu(0.)44 b(This)33 b(implies)h(that)f(\012)2477 1692 y Fl(\025Q)2606 1677 y Fn(!)27 b Fu(\012)33 b(w)m(eakly)h(as)f Fo(\025)28 b Fn(#)f Fu(0)33 b(and)552 1888 y Fn(k)p Fu(\012)p Fn(k)28 b(\024)g Fu(lim)17 b(inf)934 1951 y Fl(\025)p Fw(#)p Fk(0)1141 1888 y Fn(k)p Fu(\012)1261 1903 y Fl(\025Q)1363 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y Fu(\()p Fo(A)p Fu(\))28 b(=)f(e)1427 2887 y Fk(i)p Fl(tL)1520 2898 y Fb(Q)1577 2928 y Fo(A)p Fu(e)1693 2887 y Fw(\000)p Fk(i)p Fl(tL)1841 2898 y Fb(Q)1898 2928 y Fo(;)212 b Fu(e)2180 2887 y Fk(i)p Fl(tL)2273 2898 y Fb(Q)2330 2928 y Fn(H)2415 2887 y Fk(+)2501 2928 y Fn(\032)28 b(H)2691 2887 y Fk(+)2751 2928 y Fo(:)752 b Fu(\(5.38\))0 3099 y(Under)46 b(the)f(conditions)i (of)d(Theorem)j(5.6)d(\(recall)i(Prop)s(osition)f(2.14\),)j(the)e (second)g(relation)f(in)0 3219 y(\(5.38\))32 b(is)h(equiv)-5 b(alen)m(t)34 b(to)1658 3340 y Fo(L)1724 3355 y Fl(Q)1784 3340 y Fu(\012)1854 3355 y Fl(Q)1942 3340 y Fu(=)28 b(0)p Fo(:)1408 b Fu(\(5.39\))0 3511 y(In)36 b(this)g(section)g(w)m(e)g(giv)m (e)h(an)e(elemen)m(tary)i(direct)f(pro)s(of)f(of)g(\(5.39\).)51 b(This)36 b(v)m(eri\014es)i(that)d Fo(L)3448 3526 y Fl(Q)3543 3511 y Fu(is)h(the)0 3631 y(Liouvillean)e(of)e Fo(\034)656 3646 y Fl(Q)748 3631 y Fu(without)h(resort)g(to)g(Theorem)g(2.11.)146 3751 y(W)-8 b(e)50 b(consider)g(only)f(the)h(case)g(of)e(analytic)i(p)s (erturbations)f Fo(Q)56 b Fn(2)g Fp(M)2866 3766 y Fl(\034)2909 3751 y Fu(.)92 b(The)50 b(extension)h(to)0 3872 y(b)s(ounded)27 b Fo(Q)g Fu(and)g(un)m(b)s(ounded)g Fo(Q)g Fu(satisfying)h(conditions)f (of)f(Theorem)i(5.6)e(is)h(immediate)h(using)f(the)0 3992 y(strong)i(resolv)m(en)m(t)i(con)m(v)m(ergence)g(of)d (Liouvilleans)i(and)f(the)g(w)m(eak)h(con)m(v)m(ergence)h(of)e Fo(\014)6 b Fu(-KMS)28 b(v)m(ectors.)146 4113 y(First,)33 b(the)g(relation)1348 4233 y(e)1391 4192 y Fk(i)p Fl(t)p Fk(\()p Fl(L)p Fk(+)p Fl(Q)p Fk(\))1654 4233 y Fu(\012)1724 4248 y Fl(Q)1812 4233 y Fu(=)28 b(E)1982 4248 y Fl(Q)2042 4233 y Fu(\()p Fo(t)22 b Fu(+)g(i)p Fo(\014)6 b Fu(\)\012)0 4404 y(and)46 b(analytic)h(con)m(tin)m(uation)g(yield)g(that)f(\012) 1709 4419 y Fl(Q)1819 4404 y Fn(2)51 b(D)s Fu(\(exp)q(\(i)p Fo(z)t Fu(\()p Fo(L)32 b Fu(+)f Fo(Q)p Fu(\)\))46 b(for)f(all)h Fo(z)t Fu(,)k(and)c(so)h(\012)3603 4419 y Fl(Q)3713 4404 y Fn(2)0 4524 y(D)s Fu(\()p 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913 y(\(e)1290 872 y Fl(\014)s Fk(\()p Fl(L)p Fk(+)p Fl(Q)p Fk(\))p Fl(=)p Fk(2)1621 913 y Fu(\010)p Fn(j)p Fo(J)9 b(QJ)g Fu(\012)1992 928 y Fl(Q)2053 913 y Fu(\))27 b(=)h(\(\010)p Fn(j)p Fo(Q)p Fu(\012\))p Fo(:)0 1071 y Fu(This)34 b(relation)e(yields)422 1252 y(\(e)503 1211 y Fl(\014)s Fk(\()p Fl(L)p Fk(+)p Fl(Q)p Fk(\))p Fl(=)p Fk(2)834 1252 y Fu(\010)p Fn(j)p Fu(\()p Fo(L)23 b Fu(+)f Fo(Q)g Fn(\000)h Fo(J)9 b(QJ)g Fu(\)\012)1667 1267 y Fl(Q)1727 1252 y Fu(\))28 b(=)f(\(e)1977 1211 y Fl(\014)s Fk(\()p Fl(L)p Fk(+)p Fl(Q)p Fk(\))p Fl(=)p Fk(2)2309 1252 y Fu(\010)p Fn(j)p Fu(\()p Fo(L)22 b Fu(+)g Fo(Q)p Fu(\)\012)2816 1267 y Fl(Q)2876 1252 y Fu(\))h Fn(\000)f Fu(\(\010)p Fn(j)p Fo(Q)p Fu(\012\))1793 1468 y(=)27 b(\(\010)p Fn(j)p Fu(\()p Fo(L)c Fu(+)f Fo(Q)p Fu(\)\012\))h Fn(\000)f Fu(\(\010)p Fn(j)p Fo(Q)p Fu(\012\))1793 1684 y(=)27 b(\(\010)p Fn(j)p Fo(L)p Fu(\012\))i(=)e(0)p Fo(:)0 1882 y Fu(Since)34 b(e)298 1846 y Fl(\014)s Fk(\()p Fl(L)p Fk(+)p Fl(Q)p Fk(\))p Fl(=)p Fk(2)653 1857 y 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39 38 bop 0 407 a Fj(Pro)s(of.)61 b Fu(Note)31 b(\014rst)h(that)e Fn(f)p Fu(\012\()p Fo(z)t Fu(\))f(:)f Fo(z)k Fn(2)c Fo(Z)7 b Fn(g)31 b Fu(is)g(a)g(compact)g(subset)h(of)f Fn(H)h Fu(and)f(that)f(b)m(y)i(the)f(uniform)0 527 y(b)s(oundedness)42 b(principle)e Fo(C)47 b Fu(:=)39 b(sup)1393 551 y Fl(n)1456 527 y Fn(k)p Fo(A)1579 542 y Fl(n)1626 527 y Fn(k)g Fo(<)g Fn(1)p Fu(.)64 b(Let)39 b Fo(\017)h(>)f Fu(0)g(b)s(e)h(giv)m(en.)65 b(Then)41 b(there)f(exists)h(a)0 648 y(\014nite)33 b(dimensional)h(pro) 5 b(jection)34 b Fo(P)46 b Fu(suc)m(h)34 b(that)e(sup)1937 671 y Fl(z)s Fw(2)p Fl(Z)2093 648 y Fn(k)p Fu(\(1)22 b Fn(\000)g Fo(P)14 b Fu(\)\012\()p Fo(z)t Fu(\))p Fn(k)28 b Fo(<)g(\017)p Fu(.)43 b(Since)664 867 y Fn(k)p Fu(\()p Fo(A)825 882 y Fl(n)894 867 y Fn(\000)22 b Fo(A)p Fu(\)\012\()p Fo(z)t Fu(\))p Fn(k)84 b(\024)28 b(k)p Fu(\()p Fo(A)1699 882 y Fl(n)1768 867 y Fn(\000)23 b Fo(A)p Fu(\))p Fo(P)14 b Fn(k)j Fu(sup)2269 891 y Fl(z)s Fw(2)p Fl(Z)2425 867 y Fn(k)p Fu(\012\()p Fo(z)t Fu(\))p Fn(k)1433 1059 y Fu(+)g(sup)1673 1082 y Fl(n)1736 1059 y Fn(k)p Fo(A)1859 1074 y Fl(n)1928 1059 y Fn(\000)23 b Fo(A)p Fn(k)17 b Fu(sup)2314 1082 y Fl(z)s Fw(2)p Fl(Z)2470 1059 y Fn(k)p Fu(\(1)22 b Fn(\000)h Fo(P)14 b Fu(\)\012\()p Fo(z)t Fu(\))p Fn(k)p Fo(;)0 1277 y Fu(w)m(e)34 b(deriv)m(e)g(lim)17 b(sup)730 1301 y Fl(n)793 1277 y Fn(k)p Fu(\()p Fo(A)954 1292 y Fl(n)1023 1277 y Fn(\000)23 b Fo(A)p Fu(\)\012\()p Fo(z)t Fu(\))p Fn(k)29 b Fo(<)e Fu(2)p Fo(C)7 b(\017)p Fu(.)p Fe(2)146 1448 y Fu(The)25 b(follo)m(wing)f(prop)s(erties)g(of)g (the)g(strong)f(con)m(v)m(ergence)k(of)c(functions)i(of)e(self-adjoin)m (t)h(op)s(erators)0 1568 y(are)33 b(pro)m(v)m(en)h(e.g.)43 b(in)33 b([RS1]:)0 1796 y Fj(Prop)s(osition)38 b(A.4)48 b Fg(Supp)-5 b(ose)46 b(that)i Fo(H)1502 1811 y Fl(n)1549 1796 y Fg(,)i Fo(H)55 b Fg(ar)-5 b(e)46 b(self-adjoint)g(op)-5 b(er)g(ators.)81 b(Then)47 b(the)g(fol)5 b(lowing)0 1917 y(c)-5 b(onditions)34 b(ar)-5 b(e)34 b(e)-5 b(quivalent:)41 2110 y Fu(\(1\))48 b Fg(L)-5 b(et)36 b Fo(z)428 2125 y Fk(0)495 2110 y Fn(62)589 1999 y Fc(\020)665 2035 y(S)748 2062 y Fw(1)748 2139 y Fl(n)p Fk(=1)902 2110 y Fo(\033)t Fu(\()p Fo(H)1080 2125 y Fl(n)1127 2110 y Fu(\))1165 1999 y Fc(\021)1224 2022 y Fk(cl)1314 2110 y Fu(\()p Fg(for)f(instanc)-5 b(e,)34 b Fu(Im)p Fo(z)2080 2125 y Fk(0)2148 2110 y Fn(6)p Fu(=)27 b(0\))p Fg(.)45 b(Then)1275 2367 y Fu(s)p Fn(\000)d Fu(lim)1407 2426 y Fl(n)p Fw(!1)1591 2367 y Fu(\()p Fo(z)1674 2382 y Fk(0)1736 2367 y Fn(\000)23 b Fo(H)1917 2382 y Fl(n)1963 2367 y Fu(\))2001 2325 y Fw(\000)p Fk(1)2123 2367 y Fu(=)28 b(\()p Fo(z)2310 2382 y Fk(0)2372 2367 y Fn(\000)22 b Fo(H)8 b Fu(\))2598 2325 y Fw(\000)p Fk(1)2692 2367 y Fo(:)41 2720 y Fu(\(2\))48 b Fg(If)g Fo(f)59 b Fg(is)47 b(a)h(b)-5 b(ounde)g(d)47 b(c)-5 b(ontinuous)48 b(function)g(on)2089 2610 y Fc(\020)2165 2645 y(S)2248 2672 y Fw(1)2248 2749 y Fl(n)p Fk(=1)2402 2720 y Fo(\033)t Fu(\()p Fo(H)2580 2735 y Fl(n)2627 2720 y Fu(\))2665 2610 y Fc(\021)2724 2632 y Fk(cl)2780 2720 y Fg(,)j(then)c Fo(f)11 b Fu(\()p Fo(H)3268 2735 y Fl(n)3315 2720 y Fu(\))52 b Fn(!)f Fo(f)11 b Fu(\()p Fo(H)d Fu(\))214 2862 y Fg(str)-5 b(ongly.)146 3122 y Fu(Note)40 b(that)f(\(1\))g(in)g (the)h(ab)s(o)m(v)m(e)g(prop)s(osition)f(holds)h(for)f(an)m(y)h(c)m (hoice)g(of)f Fo(z)2936 3137 y Fk(0)3015 3122 y Fu(if)g(it)h(holds)f (for)g(one)0 3242 y(c)m(hoice)34 b(of)e Fo(z)446 3257 y Fk(0)486 3242 y Fu(.)146 3363 y(If)g(the)g(conditions)h(of)e(ab)s(o)m (v)m(e)i(prop)s(osition)f(are)f(satis\014ed)i(w)m(e)g(sa)m(y)g(that)e Fo(H)2911 3378 y Fl(n)2986 3363 y Fn(!)c Fo(H)39 b Fu(in)32 b(the)g(strong)0 3483 y(resolv)m(en)m(t)j(sense.)0 3711 y Fj(Prop)s(osition)j(A.5)48 b Fg(Supp)-5 b(ose)41 b(that)g Fo(H)1490 3726 y Fl(n)1537 3711 y Fg(,)i Fo(H)49 b Fg(ar)-5 b(e)41 b(self-adjoint)f(op)-5 b(er)g(ators,)42 b Fo(H)49 b Fg(is)41 b(essential)5 b(ly)40 b(self-)0 3832 y(adjoint)h(on)f Fn(D)k Fg(and)d Fu(lim)935 3847 y Fl(n)999 3832 y Fo(H)1080 3847 y Fl(n)1126 3832 y Fu(\011)f(=)f Fo(H)8 b Fu(\011)40 b Fg(for)h Fu(\011)e Fn(2)h(D)s Fg(.)63 b(Then)41 b Fo(H)2460 3847 y Fl(n)2546 3832 y Fn(!)e Fo(H)48 b Fg(in)41 b(the)g(str)-5 b(ong)41 b(r)-5 b(esolvent)0 3952 y(sense.)0 4180 y Fj(Pro)s(of.)65 b Fu(Let)33 b(Im)p Fo(z)g Fn(6)p Fu(=)27 b(0.)43 b(Then)34 b(\()p Fo(z)27 b Fn(\000)c Fo(H)8 b Fu(\))p Fn(D)29 b Fu(=:)f Fn(D)1865 4195 y Fk(1)1937 4180 y Fu(is)33 b(dense)h(in)f Fn(H)q Fu(.)43 b(F)-8 b(or)32 b(\011)c Fn(2)g(D)3020 4195 y Fk(1)3059 4180 y Fu(,)427 4400 y(\()p Fo(z)f Fn(\000)c Fo(H)8 b Fu(\))764 4359 y Fw(\000)p Fk(1)858 4400 y Fu(\011)22 b Fn(\000)g Fu(\()p Fo(z)27 b Fn(\000)c Fo(H)1346 4415 y Fl(n)1392 4400 y Fu(\))1430 4359 y Fw(\000)p Fk(1)1525 4400 y Fu(\011)k(=)h(\()p Fo(z)f Fn(\000)22 b Fo(H)2022 4415 y Fl(n)2069 4400 y Fu(\))2107 4359 y Fw(\000)p Fk(1)2201 4400 y Fu(\()p Fo(H)30 b Fn(\000)23 b Fo(H)2531 4415 y Fl(n)2577 4400 y Fu(\)\()p Fo(z)k Fn(\000)c Fo(H)8 b Fu(\))2952 4359 y Fw(\000)p Fk(1)3045 4400 y Fu(\011)28 b Fn(!)f Fu(0)p Fo(:)0 4620 y Fe(2)146 4791 y Fu(The)34 b(follo)m(wing)f(prop)s(osition)f(pla)m(ys)i(an)f(imp)s(ortan)m(t)f (role)h(in)g(sev)m(eral)h(argumen)m(ts)g(in)f(our)f(pap)s(er.)0 5019 y Fj(Prop)s(osition)38 b(A.6)48 b Fg(Supp)-5 b(ose)44 b(that)g Fo(H)1496 5034 y Fl(n)1543 5019 y Fg(,)i Fo(H)52 b Fg(ar)-5 b(e)44 b(self-adjoint)f(op)-5 b(er)g(ators)44 b(and)f Fo(H)3154 5034 y Fl(n)3246 5019 y Fn(!)i Fo(H)52 b Fg(in)44 b(the)0 5139 y(str)-5 b(ong)34 b(r)-5 b(esolvent)33 b(sense.)44 b(Supp)-5 b(ose)34 b(that)g Fu(\012)1636 5154 y Fl(n)1684 5139 y Fo(;)17 b Fu(\012)27 b Fn(2)h(H)36 b Fg(such)e(that)g Fu(\012)2528 5154 y Fl(n)2603 5139 y Fn(!)28 b Fu(\012)34 b Fg(we)-5 b(akly)34 b(and)g Fn(k)p Fo(H)3461 5154 y Fl(n)3507 5139 y Fu(\012)3577 5154 y Fl(n)3625 5139 y Fn(k)27 b(\024)0 5260 y Fo(C)7 b Fg(.)45 b(Then)34 b Fu(\012)28 b Fn(2)g(D)s Fu(\()p Fo(H)8 b Fu(\))p Fg(,)34 b Fu(w)q Fn(\000)18 b Fu(lim)1208 5275 y Fl(n)1272 5260 y Fo(H)1353 5275 y Fl(n)1399 5260 y Fu(\012)1469 5275 y Fl(n)1552 5260 y Fg(exists)34 b(and)h Fo(H)8 b Fu(\012)27 b(=)h(w)q Fn(\000)18 b Fu(lim)2599 5275 y Fl(n)2663 5260 y Fo(H)2744 5275 y Fl(n)2790 5260 y Fu(\012)2860 5275 y Fl(n)2908 5260 y Fg(.)1841 5753 y Fu(39)p eop %%Page: 40 40 40 39 bop 0 407 a Fj(Remark.)76 b Fu(By)44 b(the)g(uniform)f(b)s (oundedness)j(principle,)h(the)d(condition)g Fn(k)p Fo(H)2987 422 y Fl(n)3033 407 y Fu(\012)3103 422 y Fl(n)3151 407 y Fn(k)h(\024)i Fo(C)j Fu(can)43 b(b)s(e)0 527 y(replaced)34 b(b)m(y)f(the)g(existence)i(of)d(w)q Fn(\000)18 b Fu(lim)f Fo(H)1613 542 y Fl(n)1660 527 y Fu(\012)1730 542 y Fl(n)1778 527 y Fu(.)0 648 y Fj(Pro)s(of.)62 b Fu(Since)32 b(the)f(ball)g(of)g (radius)h Fo(C)38 b Fu(in)31 b(a)f(Hilb)s(ert)i(space)g(is)f(w)m(eakly) i(sequen)m(tially)h(compact,)e(one)0 768 y(can)h(\014nd)g(a)f(w)m (eakly)j(con)m(v)m(ergen)m(t)f(subsequence)j Fo(H)1898 783 y Fl(n)1941 795 y Fb(k)1982 768 y Fu(\012)2052 783 y Fl(n)2095 795 y Fb(k)2138 768 y Fu(.)43 b(Set)33 b(\011)28 b(=)g(w)c Fn(\000)e Fu(lim)2913 783 y Fl(k)r Fw(!1)3113 768 y Fo(H)3194 783 y Fl(n)3237 795 y Fb(k)3279 768 y Fu(\012)3349 783 y Fl(n)3392 795 y Fb(k)3434 768 y Fu(.)146 888 y(Let)33 b Fn(D)d Fu(:=)579 814 y Fc(S)559 973 y Fl(r)r(>)p Fk(0)700 888 y Fu(Ran)o(1)923 904 y Fk([)p Fw(\000)p Fl(r)n(;r)r Fk(])1105 888 y Fu(\()p Fo(H)8 b Fu(\).)43 b(Let)32 b(\010)c Fn(2)g(D)36 b Fu(and)c Fo(f)39 b Fn(2)28 b Fo(C)2266 852 y Fw(1)2259 913 y Fk(0)2340 888 y Fu(\()p Fm(R)p Fu(\))33 b(suc)m(h)h(that)e Fo(f)11 b Fu(\()p Fo(H)d Fu(\)\010)27 b(=)h(\010.)44 b(Then)1310 1169 y(\010)28 b(=)g Fo(f)11 b Fu(\()p Fo(H)d Fu(\)\010)27 b(=)52 b(lim)1936 1229 y Fl(n)p Fw(!1)2137 1169 y Fo(f)11 b Fu(\()p Fo(H)2315 1184 y Fl(n)2361 1169 y Fu(\)\010)1144 1421 y Fo(H)d Fu(\010)28 b(=)f Fo(f)11 b Fu(\()p Fo(H)d Fu(\))p Fo(H)g Fu(\010)27 b(=)52 b(lim)1947 1481 y Fl(n)p Fw(!1)2148 1421 y Fo(f)11 b Fu(\()p Fo(H)2326 1436 y Fl(n)2372 1421 y Fu(\))p Fo(H)2491 1436 y Fl(n)2538 1421 y Fu(\010)p Fo(;)0 1622 y Fu(and)1170 1814 y(\()p Fo(H)d Fu(\010)p Fn(j)p Fu(\012\))28 b(=)49 b(lim)1634 1877 y Fl(k)r Fw(!1)1814 1814 y Fu(\()p Fo(H)1933 1829 y Fl(n)1976 1841 y Fb(k)2018 1814 y Fo(f)11 b Fu(\()p Fo(H)2196 1829 y Fl(n)2239 1841 y Fb(k)2280 1814 y Fu(\)\010)p Fn(j)p Fu(\012)2486 1829 y Fl(n)2529 1841 y Fb(k)2572 1814 y Fu(\))1531 2075 y(=)49 b(lim)1634 2138 y Fl(k)r Fw(!1)1814 2075 y Fu(\()p Fo(f)11 b Fu(\()p Fo(H)2030 2090 y Fl(n)2073 2102 y Fb(k)2114 2075 y Fu(\)\010)p Fn(j)p Fo(H)2331 2090 y Fl(n)2374 2102 y Fb(k)2416 2075 y Fu(\012)2486 2090 y Fl(n)2529 2102 y Fb(k)2572 2075 y Fu(\))1531 2336 y(=)27 b(\(\010)p Fn(j)p Fu(\011\))p Fo(:)3506 2077 y Fu(\(A.40\))0 2555 y(Since)34 b Fn(D)h Fu(is)e(a)f(core)h(for)f Fo(H)8 b Fu(,)32 b(\012)c Fn(2)h(D)s Fu(\()p Fo(H)8 b Fu(\))31 b(and)i Fo(H)8 b Fu(\012)27 b(=)h(\011)146 2675 y(No)m(w)50 b(assume)g(that)f(w)q Fn(\000)18 b Fu(lim)1267 2690 y Fl(n)1331 2675 y Fo(H)1412 2690 y Fl(n)1458 2675 y Fu(\012)50 b(do)s(es)f(not)g(exist.)93 b(Then)50 b(there)g(exists)g (\010)56 b Fn(2)g(H)50 b Fu(and)f(a)0 2796 y(subsequence)36 b Fo(H)632 2811 y Fl(n)675 2823 y Fb(k)717 2796 y Fu(\012)d(suc)m(h)h (that)1240 3015 y Fn(j)p Fu(\(\010)p Fn(j)p Fo(H)1485 3030 y Fl(n)1528 3042 y Fb(k)1570 3015 y Fu(\012\))22 b Fn(\000)h Fu(\(\010)p Fn(j)p Fo(H)8 b Fu(\012\))p Fn(j)27 b(\025)h Fo(\017)g(>)g Fu(0)p Fo(:)966 b Fu(\(A.41\))0 3235 y(Using)42 b(again)f(the)g(w)m(eak)i(sequen)m(tial)g(compactness)h (of)d(the)g(ball)h(of)e(radius)i Fo(C)48 b Fu(and)42 b(passing)g(to)f(a)0 3355 y(subsubsequence)e(w)m(e)d(ma)m(y)g(assume)g (that)f(w)q Fn(\000)18 b Fu(lim)1914 3370 y Fl(k)r Fw(!1)2115 3355 y Fo(H)2196 3370 y Fl(n)2239 3382 y Fb(k)2281 3355 y Fu(\012)35 b(exists.)52 b(Rep)s(eating)35 b(the)g(argumen)m(ts)0 3476 y(of)d(\(A.40\),)g(w)m(e)i(see)g(that)e(w)q Fn(\000)18 b Fu(lim)1258 3491 y Fl(k)r Fw(!1)1459 3476 y Fo(H)1540 3491 y Fl(n)1583 3503 y Fb(k)1625 3476 y Fu(\012)28 b(=)f Fo(H)8 b Fu(\012.)44 b(This)34 b(con)m(tradicts)f(\(A.41\).)44 b Fe(2)0 3814 y Fs(A.2)135 b(An)44 b(in)l(terp)t(olation)j(theorem)0 3999 y Fu(V)-8 b(arious)33 b(v)m(ersions)i(of)d(the)h(follo)m(wing)h (in)m(terp)s(olation)f(theorem)g(for)f(linear)i(op)s(erators)e(can)h(b) s(e)g(found)0 4119 y(throughout)f(literature,)i(see)f(e.g.)44 b([OP])33 b(\(where)h(a)e(di\013eren)m(t)i(pro)s(of)e(is)h(outlined\))g (and)g([RS2)o(].)0 4347 y Fj(Theorem)75 b(A.7)49 b Fg(L)-5 b(et)28 b Fn(H)978 4362 y Fk(1)1018 4347 y Fo(;)17 b Fn(H)1146 4362 y Fk(2)1213 4347 y Fg(b)-5 b(e)27 b(Hilb)-5 b(ert)28 b(sp)-5 b(ac)g(es)27 b(and)g(let)h Fo(H)2318 4362 y Fl(i)2374 4347 y Fg(b)-5 b(e)27 b(a)g(p)-5 b(ositive)27 b(\(p)-5 b(ossibly)27 b(unb)-5 b(ounde)g(d\))0 4467 y(op)g(er)g(ator)34 b(on)g Fn(H)606 4482 y Fl(i)634 4467 y Fg(.)45 b(L)-5 b(et)34 b Fn(D)953 4482 y Fk(1)1027 4467 y Fg(b)-5 b(e)34 b(a)g(c)-5 b(or)g(e)33 b(of)h Fo(H)1629 4482 y Fk(1)1669 4467 y Fg(.)44 b(L)-5 b(et)35 b Fo(T)41 b Fn(2)28 b(B)s Fu(\()p Fn(H)2293 4482 y Fk(1)2333 4467 y Fo(;)17 b Fn(H)2461 4482 y Fk(2)2500 4467 y Fu(\))35 b Fg(with)f Fn(k)p Fo(T)14 b Fn(k)27 b Fu(=)g Fo(c)3127 4482 y Fk(0)3201 4467 y Fg(b)-5 b(e)34 b(such)g(that:)0 4588 y Fu(\(a\))h Fo(T)14 b Fn(D)308 4603 y Fk(1)374 4588 y Fn(\032)28 b(D)s Fu(\()p Fo(H)678 4603 y Fk(2)717 4588 y Fu(\))p Fg(.)0 4708 y Fu(\(b\))35 b Fg(F)-7 b(or)34 b Fu(\011)27 b Fn(2)h(D)621 4723 y Fk(1)661 4708 y Fg(,)35 b Fn(k)p Fo(H)857 4723 y Fk(2)896 4708 y Fo(T)14 b Fu(\011)p Fn(k)27 b(\024)h Fo(c)1267 4723 y Fk(1)1306 4708 y Fn(k)p Fo(H)1437 4723 y Fk(1)1476 4708 y Fu(\011)p Fn(k)p Fg(.)0 4829 y(Then,)34 b(for)h(any)f Fu(0)28 b Fn(\024)g Fo(\025)g Fn(\024)g Fu(1)p Fg(,)34 b Fo(T)14 b Fn(D)s Fu(\()p Fo(H)1389 4792 y Fl(\025)1381 4853 y Fk(1)1433 4829 y Fu(\))28 b Fn(\032)g(D)s Fu(\()p Fo(H)1811 4792 y Fl(\025)1803 4853 y Fk(2)1856 4829 y Fu(\))34 b Fg(and)h(for)f Fu(\011)28 b Fn(2)g(D)s Fu(\()p Fo(H)2678 4792 y Fl(\025)2670 4853 y Fk(1)2722 4829 y Fu(\))p Fg(,)1333 5048 y Fn(k)p Fo(H)1472 5007 y Fl(\025)1464 5073 y Fk(2)1516 5048 y Fo(T)14 b Fu(\011)p Fn(k)27 b(\024)i Fo(c)1888 5007 y Fk(1)p Fw(\000)p Fl(\025)1888 5073 y Fk(0)2023 5048 y Fo(c)2065 5007 y Fl(\025)2065 5073 y Fk(1)2110 5048 y Fn(k)p Fo(H)2249 5007 y Fl(\025)2241 5073 y Fk(1)2294 5048 y Fu(\011)p Fn(k)p Fo(:)1059 b Fu(\(A.42\))1841 5753 y(40)p eop %%Page: 41 41 41 40 bop 0 407 a Fj(Pro)s(of.)65 b Fu(Clearly)-8 b(,)34 b(w)m(e)f(ma)m(y)h(assume)g(that)e Fo(c)1671 422 y Fk(0)1738 407 y Fu(=)c Fo(c)1884 422 y Fk(1)1951 407 y Fu(=)f(1.)146 527 y(First)33 b(let)g(us)g(sho)m(w)h(that)e Fo(T)14 b Fn(D)s Fu(\()p Fo(H)1373 542 y Fk(1)1412 527 y Fu(\))27 b Fn(\032)h(D)s Fu(\()p Fo(H)1781 542 y Fk(2)1820 527 y Fu(\))33 b(and)1188 747 y Fn(k)p Fo(H)1319 762 y Fk(2)1358 747 y Fo(T)14 b Fu(\011)p Fn(k)27 b(\024)h(k)p Fo(H)1818 762 y Fk(1)1857 747 y Fu(\011)p Fn(k)p Fo(;)81 b Fu(\011)28 b Fn(2)g(D)s Fu(\()p Fo(H)2488 762 y Fk(1)2527 747 y Fu(\))p Fo(:)914 b Fu(\(A.43\))0 967 y(Let)33 b(\011)27 b Fn(2)h(D)s Fu(\()p Fo(H)571 982 y Fk(1)610 967 y Fu(\).)44 b(Then)33 b(there)h(exist)f(\011)1529 982 y Fl(n)1604 967 y Fn(2)28 b(D)1775 982 y Fk(1)1847 967 y Fu(suc)m(h)34 b(that)e(\011)2354 982 y Fl(n)2429 967 y Fn(!)27 b Fu(\011)32 b(and)h Fo(H)2935 982 y Fk(1)2974 967 y Fu(\011)3050 982 y Fl(n)3125 967 y Fn(!)27 b Fo(H)3333 982 y Fk(1)3372 967 y Fu(\011.)44 b(No)m(w)1056 1187 y Fn(k)p Fo(H)1187 1202 y Fk(2)1226 1187 y Fu(\()p Fo(T)14 b Fu(\011)1411 1202 y Fl(n)1480 1187 y Fn(\000)23 b Fo(T)14 b Fu(\011)1727 1202 y Fl(m)1793 1187 y Fu(\))p Fn(k)27 b(\024)h(k)p Fo(H)2144 1202 y Fk(1)2183 1187 y Fu(\(\011)2297 1202 y Fl(n)2366 1187 y Fn(\000)23 b Fu(\011)2542 1202 y Fl(m)2608 1187 y Fu(\))p Fn(k)p Fo(:)0 1407 y Fu(Th)m(us)41 b Fo(H)335 1422 y Fk(2)374 1407 y Fo(T)14 b Fu(\011)521 1422 y Fl(n)607 1407 y Fu(is)40 b(Cauc)m(h)m(y)-8 b(,)42 b(hence)f(con)m(v)m(ergen)m (t.)66 b Fo(T)14 b Fu(\011)2059 1422 y Fl(n)2144 1407 y Fu(is)40 b(ob)m(viously)h(con)m(v)m(ergen)m(t.)66 b Fo(H)3314 1422 y Fk(2)3392 1407 y Fu(is)40 b(closed.)0 1528 y(Hence)34 b Fo(T)14 b Fu(\011)27 b Fn(2)h(D)s Fu(\()p Fo(H)757 1543 y Fk(2)796 1528 y Fu(\).)43 b(\(A.43\))33 b(follo)m(ws)g(b)m(y)g(passing)h(to)e(the)h(limit)g(in)g Fn(k)p Fo(H)2774 1543 y Fk(2)2813 1528 y Fo(T)14 b Fu(\011)2960 1543 y Fl(n)3007 1528 y Fn(k)27 b(\024)h(k)p Fo(H)3320 1543 y Fk(1)3359 1528 y Fu(\011)3435 1543 y Fl(n)3482 1528 y Fn(k)p Fu(.)146 1648 y(Let)33 b(\010)28 b Fn(2)g(D)s Fu(\()p Fo(H)712 1663 y Fk(2)751 1648 y Fu(\),)33 b(\012)28 b Fn(2)g(H)1125 1663 y Fk(1)1197 1648 y Fu(and)33 b Fo(\017)28 b(>)f Fu(0.)43 b(F)-8 b(or)32 b(0)c Fn(\024)g Fu(Re)p Fo(z)k Fn(\024)d Fu(1)j(set)1239 1868 y Fo(F)1302 1883 y Fl(\017)1335 1868 y Fu(\()p Fo(z)t Fu(\))d(:=)e(\()p Fo(H)p 1746 1789 40 3 v 1746 1827 a Fl(z)1738 1893 y Fk(2)1785 1868 y Fu(\010)p Fn(j)p Fo(T)14 b Fu(\()p Fo(H)2073 1883 y Fk(1)2135 1868 y Fu(+)22 b Fo(\017)p Fu(\))2310 1827 y Fw(\000)p Fl(z)2405 1868 y Fu(\012\))p Fo(:)0 2088 y(F)63 2103 y Fl(\017)96 2088 y Fu(\()p Fo(z)t Fu(\))33 b(is)g(a)f(con)m(tin)m(uous)j(function)e(in)g(the)g(strip)g(0)27 b Fn(\024)h Fu(Re)p Fo(z)33 b Fn(\024)28 b Fu(1,)k(analytic)i(in)e(its) h(in)m(terior,)h(and)1252 2308 y Fn(j)p Fo(F)1343 2323 y Fl(\017)1375 2308 y Fu(\()p Fo(z)t Fu(\))p Fn(j)28 b(\024)h(k)p Fu(\()p Fo(H)1831 2323 y Fk(2)1892 2308 y Fu(+)22 b(1\)\010)p Fn(k)p Fo(\017)2236 2267 y Fw(\000)p Fk(1)2330 2308 y Fn(k)p Fu(\012)p Fn(k)p Fo(:)146 2528 y Fu(F)-8 b(or)32 b(Re)p Fo(z)h Fu(=)27 b(0)1501 2648 y Fn(j)p Fo(F)1592 2663 y Fl(\017)1625 2648 y Fu(\()p Fo(z)t Fu(\))p Fn(j)h(\024)g(k)p Fu(\010)p Fn(kk)p Fu(\012)p Fn(k)p Fo(:)0 2823 y Fu(F)-8 b(or)32 b(Re)p Fo(z)g Fu(=)c(1,)k(\()p Fo(H)698 2838 y Fk(1)760 2823 y Fu(+)22 b Fo(\017)p Fu(\))935 2787 y Fw(\000)p Fl(z)1030 2823 y Fu(\012)28 b Fn(2)g(D)s Fu(\()p Fo(H)1421 2838 y Fk(1)1460 2823 y Fu(\),)k(and)980 3040 y Fn(j)p Fo(F)1071 3055 y Fl(\017)1103 3040 y Fu(\()p Fo(z)t Fu(\))p Fn(j)c(\024)g(k)p Fu(\010)p Fn(kk)p Fo(H)1690 3055 y Fk(2)1729 3040 y Fo(T)14 b Fu(\()p Fo(H)1919 3055 y Fk(1)1980 3040 y Fu(+)22 b Fo(\017)p Fu(\))2155 2999 y Fw(\000)p Fl(z)2250 3040 y Fu(\012)p Fn(k)1284 3256 y(\024)28 b(k)p Fu(\010)p Fn(kk)p Fo(H)1690 3271 y Fk(1)1729 3256 y Fu(\()p Fo(H)1848 3271 y Fk(1)1910 3256 y Fu(+)22 b Fo(\017)p Fu(\))2085 3215 y Fw(\000)p Fl(z)2180 3256 y Fu(\012)p Fn(k)28 b(\024)g(k)p Fu(\010)p Fn(kk)p Fu(\012)p Fn(k)p Fo(:)0 3469 y Fu(These)34 b(estimates)h(and)d(the)h(three-line)h (theorem)f(yield)h(that)e(for)g(0)c Fn(\024)g Fo(\025)f Fn(\024)i Fu(1)1498 3689 y Fn(j)p Fo(F)1589 3704 y Fl(\017)1621 3689 y Fu(\()p Fo(\025)p Fu(\))p Fn(j)e(\024)i(k)p Fu(\010)p Fn(kk)p Fu(\012)p Fn(k)p Fo(:)0 3909 y Fu(Therefore,)34 b(for)e(\012)c Fn(2)g(H)894 3924 y Fk(1)934 3909 y Fu(,)1309 4030 y Fn(k)p Fo(H)1448 3988 y Fl(\025)1440 4054 y Fk(2)1493 4030 y Fo(T)14 b Fu(\()p Fo(H)1683 4045 y Fk(1)1744 4030 y Fu(+)22 b Fo(\017)p Fu(\))1919 3988 y Fw(\000)p Fl(\025)2020 4030 y Fu(\012)p Fn(k)28 b(\024)g(k)p Fu(\012)p Fn(k)p Fo(;)0 4204 y Fu(and)33 b(for)f(\011)27 b Fn(2)h(D)s Fu(\()p Fo(H)743 4168 y Fl(\025)735 4229 y Fk(1)788 4204 y Fu(\))940 4424 y Fn(k)p Fo(H)1079 4383 y Fl(\025)1071 4449 y Fk(2)1123 4424 y Fo(T)14 b Fu(\011)p Fn(k)27 b Fu(=)h(lim)1469 4487 y Fl(\017)p Fw(#)p Fk(0)1603 4424 y Fn(k)p Fo(H)1742 4383 y Fl(\025)1734 4449 y Fk(2)1787 4424 y Fo(T)14 b Fu(\()p Fo(H)1977 4439 y Fk(1)2038 4424 y Fu(+)22 b Fo(\017)p Fu(\))2213 4383 y Fw(\000)p Fl(\025)2313 4424 y Fu(\()p Fo(H)2432 4439 y Fk(1)2494 4424 y Fu(+)g Fo(\017)p Fu(\))2669 4383 y Fl(\025)2714 4424 y Fu(\011)p Fn(k)1347 4701 y(\024)29 b Fu(lim)1471 4764 y Fl(\017)p Fw(#)p Fk(0)1605 4701 y Fn(k)p Fu(\()p Fo(H)1774 4716 y Fk(1)1835 4701 y Fu(+)22 b Fo(\017)p Fu(\))2010 4660 y Fl(\025)2055 4701 y Fu(\011)p Fn(k)1347 4978 y Fu(=)28 b Fn(k)p Fo(H)1590 4937 y Fl(\025)1582 5003 y Fk(1)1635 4978 y Fu(\011)p Fn(k)p Fo(:)0 5166 y Fe(2)1841 5753 y Fu(41)p eop %%Page: 42 42 42 41 bop 0 407 a Ft(B)161 b(P)l(auli-Fierz)53 b(systems)0 655 y Fs(B.1)135 b(In)l(tro)t(duction)0 840 y Fu(A)31 b(large)g(part)f(of)h(the)g(motiv)-5 b(ation)31 b(for)f(the)i (formalism)f(and)g(the)g(results)i(of)d(our)h(pap)s(er)g(comes)h(from)0 960 y(quan)m(tum)45 b(statistical)g(ph)m(ysics.)81 b(A)44 b(detailed)h(description)g(of)f(their)h(application)f(to)g(P)m (auli-Fierz)0 1080 y(systems|a)30 b(certain)f(class)g(of)e(ph)m (ysically)k(motiv)-5 b(ated)28 b Fo(W)2163 1044 y Fw(\003)2203 1080 y Fu(-dynamical)g(systems|can)j(b)s(e)d(found)g(in)0 1201 y([DJ2)o(].)44 b(In)33 b(this)g(app)s(endix)h(w)m(e)f(brie\015y)h (describ)s(e)g(these)g(applications.)146 1321 y(P)m(auli-Fierz)i (systems)h(describ)s(e)g(a)d(small)i(quan)m(tum)g(system)h(\(an)e(atom) g(or)f(a)h(molecule\))h(in)m(ter-)0 1442 y(acting)e(with)g(a)g(large)g (b)s(osonic)g(reserv)m(oir.)49 b(They)36 b(arise)e(as)g(an)g(appro)m (ximation)h(to)e(non-relativistic)0 1562 y(QED)c(\(see)g(e.g.)43 b([DJ1)o(,)29 b(RZ]\),)g(and)g(they)h(ha)m(v)m(e)g(b)s(een)g(widely)g (used)g(in)f(ph)m(ysics)j(literature)d(as)g(a)g(basic)0 1682 y(paradigm)k(of)f(an)g(op)s(en)h(quan)m(tum)h(system)g([LCD,)f(W) -8 b(e].)146 1803 y(W)g(e)46 b(are)g(in)m(terested)i(in)e(the)g(case)g (where)h(the)f(radiation)g(densit)m(y)h(of)f(the)g(b)s(osonic)g(reserv) m(oir)0 1923 y(is)d(not)f(zero)h(\(in)f(particular,)k(the)c(reserv)m (oir)i(is)f(not)f(at)h(zero)f(temp)s(erature\).)74 b(F)-8 b(or)42 b(example,)k(the)0 2043 y(radiation)33 b(densit)m(y)j(can)e(b)s (e)f(giv)m(en)i(b)m(y)f(the)g(Planc)m(k)h(la)m(w)f(at)g(the)g(in)m(v)m (erse)h(temp)s(erature)g Fo(\014)g(<)29 b Fn(1)p Fu(,)k(see)0 2164 y(\(B.46\))41 b(b)s(elo)m(w.)71 b(This)43 b(corresp)s(onds)g(to)e (the)h(case)g(of)f(b)s(osons)h(in)g(thermal)g(equilibrium.)73 b(W)-8 b(e)41 b(are)0 2284 y(also)32 b(in)m(terested)i(in)f(situations) g(outside)g(thermal)f(equilibrium.)46 b(F)-8 b(or)31 b(example,)j(the)e(reserv)m(oir)i(ma)m(y)0 2405 y(consist)g(of)e(sev)m (eral)i(subreserv)m(oirs)i(at)c(distinct)i(temp)s(eratures.)146 2525 y Fo(W)252 2489 y Fw(\003)291 2525 y Fu(-dynamical)39 b(systems)h(pro)m(vide)g(a)d(natural)h(framew)m(ork)h(to)f(describ)s(e) h(P)m(auli-Fierz)g(systems)0 2645 y(with)33 b(non-zero)g(radiation)f (densit)m(y)-8 b(,)35 b(as)d(it)h(will)g(b)s(e)g(sk)m(etc)m(hed)i(b)s (elo)m(w.)0 2931 y Fs(B.2)135 b(Bose)45 b(gas)g(at)h(densit)l(y)f Fr(\032)p Fs(|Araki-W)-11 b(o)t(o)t(ds)46 b(algebras)0 3116 y Fu(If)d Fn(Z)51 b Fu(is)44 b(a)f(Hilb)s(ert)g(space,)k(then)d(w) m(e)g(will)g(write)g(\000)1982 3131 y Fk(s)2014 3116 y Fu(\()p Fn(Z)8 b Fu(\))43 b(for)f(the)i(b)s(osonic)f(F)-8 b(o)s(c)m(k)44 b(space)g(o)m(v)m(er)g(the)0 3236 y(1-particle)33 b(space)g Fn(Z)8 b Fu(.)44 b(\012)33 b(will)g(denote)g(the)g(v)-5 b(acuum)34 b(v)m(ector.)146 3357 y(F)-8 b(or)35 b(de\014niteness,)j(w)m (e)f(will)f(consider)g(the)g(Bose)g(gas)f(with)h(the)f(1-particle)h (space)g Fo(L)3329 3321 y Fk(2)3369 3357 y Fu(\()p Fm(R)3479 3321 y Fl(d)3519 3357 y Fu(\).)51 b(As-)0 3477 y(sume)28 b(that)f Fm(R)522 3441 y Fl(d)590 3477 y Fn(3)h Fo(\030)k Fn(7!)c Fo(\032)p Fu(\()p Fo(\030)5 b Fu(\))26 b(is)i(a)f(nonnegativ)m (e)h(real)f(measurable)i(function)e(describing)i(the)e(densit)m(y)0 3598 y(of)38 b(b)s(osons)g(with)h(the)g(momen)m(tum)h Fo(\030)h Fn(2)d Fm(R)1620 3561 y Fl(d)1660 3598 y Fu(.)61 b(T)-8 b(o)38 b(describ)s(e)i(the)e(Bose)h(gas)f(at)g(densit)m(y)i Fo(\032)f Fu(one)f(uses)0 3718 y(a)c(sp)s(ecial)h(v)m(on)g(Neumann)h (algebra)e(\014rst)h(describ)s(ed)h(b)m(y)f(Araki)g(and)f(W)-8 b(o)s(o)s(ds)34 b(in)h([ArW].)49 b(It)35 b(can)f(b)s(e)0 3838 y(de\014ned)g(b)m(y)f(its)h(represen)m(tation)g(in)f(the)g(Hilb)s (ert)g(space)1254 4039 y Fn(H)1339 3998 y Fk(A)-8 b(W)1489 4039 y Fu(:=)28 b(\000)1681 4054 y Fk(s)1713 4039 y Fu(\()p Fo(L)1817 3998 y Fk(2)1857 4039 y Fu(\()p Fm(R)1967 3998 y Fl(d)2007 4039 y Fu(\))22 b Fn(\010)h Fo(L)2233 3998 y Fk(2)2272 4039 y Fu(\()p Fm(R)2382 3998 y Fl(d)2423 4039 y Fu(\)\))p Fo(:)0 4239 y Fu(W)-8 b(e)36 b(will)g(write)g Fo(a)661 4254 y Fk(l)685 4239 y Fu(\()p Fo(\030)5 b Fu(\))p Fo(;)34 b(a)921 4203 y Fw(\003)921 4265 y Fk(l)961 4239 y Fu(\()p Fo(\030)5 b Fu(\),)35 b Fo(a)1198 4254 y Fk(r)1230 4239 y Fu(\()p Fo(\030)5 b Fu(\))p Fo(;)35 b(a)1467 4203 y Fw(\003)1467 4264 y Fk(r)1507 4239 y Fu(\()p Fo(\030)5 b Fu(\))34 b(for)h(the)h(creation)f(and)h(annihilation)g(op)s(erators)f (corre-)0 4360 y(sp)s(onding)28 b(to)f(the)h(left)g(and)f(righ)m(t)h Fo(L)1334 4324 y Fk(2)1374 4360 y Fu(\()p Fm(R)1484 4324 y Fl(d)1524 4360 y Fu(\))f(resp.)43 b(W)-8 b(e)28 b(de\014ne)h(the)e (left/righ)m(t)h(Araki-W)-8 b(o)s(o)s(ds)27 b(creation)0 4480 y(and)33 b(annihillation)g(op)s(etators)1038 4681 y Fo(a)1089 4645 y Fw(\003)1089 4707 y Fl(\032;)p Fk(l)1169 4681 y Fu(\()p Fo(\030)5 b Fu(\))27 b(:=)1451 4596 y Fc(p)p 1550 4596 343 4 v 1550 4681 a Fu(1)22 b(+)g Fo(\032)p Fu(\()p Fo(\030)5 b Fu(\))p Fo(a)1944 4645 y Fw(\003)1944 4707 y Fk(l)1983 4681 y Fu(\()p Fo(\030)g Fu(\))22 b(+)2227 4596 y Fc(p)p 2326 4596 174 4 v 2326 4681 a Fo(\032)p Fu(\()p Fo(\030)5 b Fu(\))p Fo(a)2551 4696 y Fk(r)2583 4681 y Fu(\()p Fo(\030)g Fu(\))p Fo(;)1038 4881 y(a)1089 4896 y Fl(\032;)p Fk(l)1169 4881 y Fu(\()p Fo(\030)g Fu(\))27 b(:=)1451 4796 y Fc(p)p 1550 4796 343 4 v 1550 4881 a Fu(1)22 b(+)g Fo(\032)p Fu(\()p Fo(\030)5 b Fu(\))p Fo(a)1944 4896 y Fk(l)1968 4881 y Fu(\()p Fo(\030)g Fu(\))21 b(+)2211 4796 y Fc(p)p 2311 4796 174 4 v 85 x Fo(\032)p Fu(\()p Fo(\030)5 b Fu(\))o Fo(a)2535 4845 y Fw(\003)2535 4906 y Fk(r)2575 4881 y Fu(\()p Fo(\030)g Fu(\))p Fo(;)1038 5081 y(a)1089 5045 y Fw(\003)1089 5106 y Fl(\032;)p Fk(r)1177 5081 y Fu(\()p Fo(\030)g Fu(\))27 b(:=)1459 4996 y Fc(p)p 1558 4996 V 1558 5081 a Fo(\032)p Fu(\()p Fo(\030)5 b Fu(\))p Fo(a)1783 5096 y Fk(l)1807 5081 y Fu(\()p Fo(\030)g Fu(\))21 b(+)2050 4996 y Fc(p)p 2150 4996 343 4 v 85 x Fu(1)h(+)g Fo(\032)p Fu(\()p Fo(\030)5 b Fu(\))o Fo(a)2543 5045 y Fw(\003)2543 5106 y Fk(r)2583 5081 y Fu(\()p Fo(\030)g Fu(\))p Fo(;)1038 5282 y(a)1089 5297 y Fl(\032;)p Fk(r)1177 5282 y Fu(\()p Fo(\030)g Fu(\))27 b(:=)1459 5196 y Fc(p)p 1558 5196 174 4 v 1558 5282 a Fo(\032)p Fu(\()p Fo(\030)5 b Fu(\))p Fo(a)1783 5245 y Fw(\003)1783 5307 y Fk(l)1822 5282 y Fu(\()p Fo(\030)g Fu(\))22 b(+)2066 5196 y Fc(p)p 2165 5196 343 4 v 2165 5282 a Fu(1)g(+)g Fo(\032)p Fu(\()p Fo(\030)5 b Fu(\))p Fo(a)2559 5297 y Fk(r)2591 5282 y Fu(\()p Fo(\030)g Fu(\))p Fo(:)1841 5753 y Fu(42)p eop %%Page: 43 43 43 42 bop 0 407 a Fu(The)23 b(left)g(Araki-W)-8 b(o)s(o)s(ds)22 b(algebra)h(is)g(denoted)g(b)m(y)h Fp(M)1941 371 y Fk(A)-8 b(W)1941 433 y Fl(\032;)p Fk(l)2085 407 y Fu(and)23 b(de\014ned)h(as)f (the)g Fo(W)2965 371 y Fw(\003)3004 407 y Fu(-algebra)f(generated)0 527 y(b)m(y)33 b(the)g(op)s(erators)1086 690 y(exp)1235 580 y Fc(\020)1294 690 y Fu(i)1338 555 y Fc(Z)1438 690 y Fu(\()p Fo(f)11 b Fu(\()p Fo(\030)5 b Fu(\))p Fo(a)1710 649 y Fw(\003)1710 715 y Fl(\032;)p Fk(l)1788 690 y Fu(\()p Fo(\030)g Fu(\))22 b(+)p 2032 609 59 4 v 22 w Fo(f)10 b Fu(\()p Fo(\030)5 b Fu(\))p Fo(a)2265 705 y Fl(\032;)p Fk(l)2344 690 y Fu(\()p Fo(\030)g Fu(\)\)d)p Fo(\030)2608 580 y Fc(\021)2667 690 y Fo(:)0 921 y Fu(Let)33 b Fo(J)238 885 y Fk(A)-8 b(W)388 921 y Fu(:=)28 b(\000\()p Fo(\017)p Fu(\),)33 b(where)g Fo(\017)g Fu(is)g(an)g(an)m(tilinear)g(in)m(v)m (olution)h(on)f Fo(L)2438 885 y Fk(2)2477 921 y Fu(\()p Fm(R)2587 885 y Fl(d)2628 921 y Fu(\))22 b Fn(\010)g Fo(L)2853 885 y Fk(2)2893 921 y Fu(\()p Fm(R)3003 885 y Fl(d)3044 921 y Fu(\))32 b(giv)m(en)i(b)m(y)1472 1141 y Fo(\017)p Fu(\()p Fo(f)1597 1156 y Fk(1)1637 1141 y Fo(;)p 1681 1060 V 17 w(f)1739 1164 y Fk(2)1779 1141 y Fu(\))27 b(:=)h(\()p Fo(f)2061 1156 y Fk(2)2101 1141 y Fo(;)p 2145 1060 V 17 w(f)2203 1164 y Fk(1)2242 1141 y Fu(\))p Fo(;)0 1361 y Fu(and)41 b(\000)f(is)i(the)f(second)h(quan)m (tization)f(functor,)i(and)e(let)g Fn(H)2284 1325 y Fk(A)-8 b(W)r Fl(;)p Fk(+)2283 1385 y Fl(\032)2522 1361 y Fu(b)s(e)41 b(the)g(closure)h(of)e(the)h(cone)g(in)0 1481 y Fn(H)85 1445 y Fk(A)-8 b(W)240 1481 y Fu(generated)34 b(b)m(y)1455 1602 y Fo(AJ)9 b(A)p Fu(\012)p Fo(;)115 b(A)28 b Fn(2)g Fp(M)2176 1560 y Fk(A)-8 b(W)2176 1626 y Fl(\032;)p Fk(l)2298 1602 y Fo(:)0 1776 y Fu(Then)43 b(\()p Fp(M)407 1740 y Fk(A)-8 b(W)407 1802 y Fl(\032;)p Fk(l)529 1776 y Fo(;)17 b Fn(H)658 1740 y Fk(A)-8 b(W)780 1776 y Fo(;)17 b(J)887 1740 y Fk(A)-8 b(W)1010 1776 y Fo(;)17 b Fn(H)1139 1740 y Fk(A)-8 b(W)q Fl(;)p Fk(+)1138 1801 y Fl(\032)1336 1776 y Fu(\))42 b(is)g(a)f(v)m(on)i(Neumann)g(algebra)f(in)g(a)f (standard)h(form.)72 b(It)41 b(de-)0 1896 y(scrib)s(es)34 b(the)f(Bose)g(gas)g(at)f(densit)m(y)j Fo(\032)p Fu(.)0 2185 y Fs(B.3)135 b(Araki-W)-11 b(o)t(o)t(ds)44 b(algebra)i(coupled)f (to)g(a)h(t)l(yp)t(e)f(I)g(factor)0 2370 y Fu(W)-8 b(e)25 b(denote)h(b)m(y)g Fn(K)g Fu(the)g(Hilb)s(ert)f(space)h(of)f(the)g (small)h(quan)m(tum)g(system.)43 b(F)-8 b(or)24 b(simplicit)m(y)-8 b(,)29 b(w)m(e)d(assume)0 2490 y(that)i(dim)17 b Fn(K)29 b Fo(<)f Fn(1)p Fu(.)41 b(W)-8 b(e)29 b(w)m(ould)g(lik)m(e)h(to)e (describ)s(e)i(the)e Fo(W)2139 2454 y Fw(\003)2178 2490 y Fu(-algebra)g(of)g(the)h(join)m(t)f(system)i(consisting)0 2611 y(of)i(the)h(small)g(system)i(with)e(the)g(algebra)g(of)f(observ) -5 b(ables)34 b Fn(B)s Fu(\()p Fn(K)q 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%%Page: 44 44 44 43 bop 0 407 a Fs(B.4)135 b(P)l(auli-Fierz)45 b Fr(W)1152 363 y Fq(\003)1198 407 y Fs(-dynamical)g(systems)0 592 y Fu(Supp)s(ose)37 b(that)f Fo(K)44 b Fu(is)36 b(a)g(self-adjoin)m(t)h (op)s(erator)f(on)g Fn(K)h Fu(describing)h(the)f(Hamiltonian)f(of)g (the)h(small)0 712 y(system.)53 b(Let)35 b Fn(j)p Fo(\030)5 b Fn(j)34 b Fu(b)s(e)h(the)h(energy)g(of)e(the)i(b)s(oson)f(of)f(momen) m(tum)j Fo(\030)5 b Fu(.)50 b(Let)35 b Fm(R)2872 676 y Fl(d)2945 712 y Fn(3)d Fo(\030)k Fn(7!)c Fo(v)t Fu(\()p Fo(\030)5 b Fu(\))31 b Fn(2)h(B)s Fu(\()p Fn(K)q Fu(\))0 832 y(describ)s(e)j(the)e(coupling)h(of)e(the)i(small)f(system)i(to)e (the)g(Bose)h(gas.)45 b(W)-8 b(e)33 b(assume)i(that)d(the)i(Bose)f(gas) 0 953 y(is)g(at)f(the)h(densit)m(y)i Fo(\032)p Fu(.)44 b(Let)32 b Fo(\025)c Fn(2)g Fm(R)p Fu(.)146 1073 y(W)-8 b(e)33 b(in)m(tro)s(duce)h(the)f(follo)m(wing)g(op)s(erators)f(on)h Fn(K)23 b(\012)g Fu(\000)2155 1088 y Fk(s)2187 1073 y Fu(\()p Fo(L)2291 1037 y Fk(2)2330 1073 y Fu(\()p Fm(R)2440 1037 y Fl(d)2481 1073 y Fu(\))f Fn(\010)g Fo(L)2706 1037 y Fk(2)2746 1073 y Fu(\()p Fm(R)2856 1037 y Fl(d)2897 1073 y Fu(\)\):)734 1341 y Fo(L)800 1300 y Fk(semi)800 1366 y(fr)970 1341 y Fu(:=)28 b Fo(K)h Fn(\012)23 b Fu(1)f(+)g(1)g Fn(\012)1652 1206 y Fc(Z)1768 1341 y Fn(j)p Fo(\030)5 b Fn(j)1872 1261 y Fc(\000)1917 1341 y Fo(a)1968 1300 y Fw(\003)1968 1366 y Fk(l)2007 1341 y Fu(\()p Fo(\030)g Fu(\))p Fo(a)2182 1356 y Fk(l)2206 1341 y Fu(\()p Fo(\030)g Fu(\))21 b Fn(\000)i Fo(a)2502 1300 y Fw(\003)2502 1366 y Fk(r)2541 1341 y Fu(\()p Fo(\030)5 b Fu(\))p Fo(a)2716 1356 y Fk(r)2748 1341 y Fu(\()p Fo(\030)g Fu(\))2872 1261 y Fc(\001)2917 1341 y Fu(d)p Fo(\030)g(;)912 1671 y(Q)989 1630 y Fk(semi)989 1695 y Fl(\032)1158 1671 y Fu(:=)1289 1535 y Fc(Z)1389 1590 y(\000)1434 1671 y Fo(v)t Fu(\()p Fo(\030)g Fu(\))21 b Fn(\012)i Fo(a)1781 1630 y Fw(\003)1781 1695 y Fl(\032;)p Fk(l)1861 1671 y Fu(\()p Fo(\030)5 b Fu(\))21 b(+)h Fo(v)2155 1630 y Fw(\003)2194 1671 y Fu(\()p Fo(\030)5 b Fu(\))22 b Fn(\012)g Fo(a)2490 1686 y Fl(\032;)p Fk(l)2570 1671 y Fu(\()p Fo(\030)5 b Fu(\))2694 1590 y Fc(\001)2739 1671 y Fu(d)p Fo(\030)g(;)0 1900 y Fu(The)35 b(op)s(erator)e Fo(L)662 1863 y Fk(semi)662 1925 y(fr)838 1900 y Fu(will)i(b)s(e)f(called)h(the)g(free)f (semi-Liouvillean.)50 b(The)35 b(full)f(semi-Liouvillean)i(for)0 2020 y(the)d(densit)m(y)i Fo(\032)d Fu(equals)1391 2140 y Fo(L)1457 2099 y Fk(semi)1457 2165 y Fl(\032)1627 2140 y Fu(:=)27 b Fo(L)1823 2099 y Fk(semi)1823 2165 y(fr)1988 2140 y Fu(+)22 b Fo(\025Q)2220 2099 y Fk(semi)2220 2165 y Fl(\032)2362 2140 y Fo(:)1121 b Fu(\(B.45\))0 2315 y(F)-8 b(or)32 b Fo(A)c Fn(2)g Fp(M)475 2330 y Fl(\032)547 2315 y Fu(w)m(e)34 b(set)1347 2444 y Fo(\034)1400 2408 y Fl(t)1389 2470 y Fk(fr)1443 2444 y Fu(\()p Fo(A)p Fu(\))27 b(:=)h(e)1793 2408 y Fk(i)p Fl(tL)1886 2384 y Ff(semi)1886 2431 y(fr)2014 2444 y Fo(A)p Fu(e)2130 2408 y Fw(\000)p Fk(i)p Fl(tL)2278 2384 y Ff(semi)2278 2431 y(fr)2405 2444 y Fo(;)1347 2645 y(\034)1400 2609 y 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(case)h(of)f(the)g(free)0 4708 y(dynamics.)0 4936 y Fj(Theorem)75 b(B.1)90 b Fu(\(1\))49 b Fg(F)-7 b(or)49 b(any)i Fo(t)p Fg(,)j Fo(\034)1515 4900 y Fl(t)1504 4962 y Fk(fr)1608 4936 y Fg(pr)-5 b(eserves)50 b(the)g(algebr)-5 b(a)50 b Fp(M)2673 4951 y Fl(\032)2763 4936 y Fg(and)g Fu(\()p Fp(M)3111 4951 y Fl(\032)3151 4936 y Fo(;)17 b(\034)3237 4951 y Fk(fr)3290 4936 y Fu(\))51 b Fg(is)f(a)h Fo(W)3706 4900 y Fw(\003)3745 4936 y Fg(-)214 5057 y(dynamic)-5 b(al)34 b(system.)41 5219 y Fu(\(2\))48 b Fo(L)280 5234 y Fk(fr)369 5219 y Fg(is)35 b(the)f(Liouvil)5 b(le)-5 b(an)34 b(for)h(the)g(dynamics)f Fo(\034)1928 5234 y Fk(fr)1981 5219 y Fg(.)1841 5753 y Fu(44)p eop %%Page: 45 45 45 44 bop 41 407 a Fu(\(3\))48 b Fg(L)-5 b(et)36 b Fo(\014)d(>)28 b Fu(0)p Fg(,)1579 527 y Fo(\032)p Fu(\()p Fo(\030)5 b Fu(\))27 b(=)h(\(e)1965 486 y Fl(\014)s Fw(j)p Fl(\030)s Fw(j)2107 527 y Fn(\000)23 b Fu(1\))2294 486 y Fw(\000)p Fk(1)2388 527 y Fo(;)1095 b Fu(\(B.46\))214 702 y Fg(and)32 b Fu(\011)477 717 y Fk(fr)558 702 y Fu(:=)27 b(e)731 665 y Fw(\000)p Fl(\014)s(K)q(=)p Fk(2)981 702 y Fn(\012)16 b Fu(\012)p Fg(.)45 b(Using)32 b(the)g(natur)-5 b(al)32 b(identi\014c)-5 b(ation)32 b(of)f Fo(l)2712 665 y Fk(2)2752 702 y Fu(\()p Fn(K)q Fu(\))i Fg(with)f Fn(K)17 b(\012)p 3333 622 78 4 v 16 w(K)r Fg(,)33 b Fu(\011)3550 717 y Fk(fr)3635 702 y Fg(c)-5 b(an)214 822 y(b)g(e)31 b(understo)-5 b(o)g(d)30 b(as)h(an)f(element)h(of)f(the)h(Hilb)-5 b(ert)31 b(sp)-5 b(ac)g(e)30 b Fn(K)15 b(\012)p 2444 742 V 13 w(K)g(\012)e Fu(\000)2686 837 y Fk(s)2719 822 y Fu(\()p Fo(L)2823 786 y Fk(2)2863 822 y Fu(\()p Fm(R)2973 786 y Fl(d)3013 822 y Fu(\))g Fn(\010)g Fo(L)3220 786 y Fk(2)3261 822 y Fu(\()p Fm(R)3371 786 y Fl(d)3411 822 y Fu(\)\))p Fg(.)43 b(Then)214 942 y Fu(\011)290 957 y Fk(fr)378 942 y Fg(is)35 b(a)g Fo(\014)6 b Fg(-KMS)34 b(ve)-5 b(ctor)35 b(for)f Fo(\034)1400 957 y Fk(fr)1454 942 y Fg(.)146 1171 y Fu(The)42 b(results)h(of)d(our)h(pap)s(er)g(are)g(the)g(main)h (tec)m(hnical)g(input)g(in)f(the)g(pro)s(of)f(of)h(the)g(follo)m(wing)0 1291 y(theorem,)33 b(whic)m(h)i(describ)s(es)f(the)f(in)m(teracting)h (dynamics:)0 1519 y Fj(Theorem)75 b(B.2)90 b Fu(\(1\))49 b Fg(Assume)34 b(that)1208 1652 y Fc(Z)1308 1788 y Fu(\(1)22 b(+)g Fn(j)p Fo(\030)5 b Fn(j)1619 1747 y Fk(2)1657 1788 y Fu(\)\(1)22 b(+)g Fo(\032)p Fu(\()p Fo(\030)5 b Fu(\)\))p Fn(k)p Fo(v)t Fu(\()p Fo(\030)g Fu(\))p Fn(k)2389 1747 y Fk(2)2426 1788 y Fu(d)p Fo(\030)33 b(<)27 b Fn(1)p Fo(:)724 b Fu(\(B.47\))214 2058 y Fg(Then)30 b(for)h(any)g Fo(t)p Fg(,)g Fo(\034)948 2022 y Fl(t)937 2083 y(\032)1009 2058 y Fg(pr)-5 b(eserves)30 b(the)h(algebr)-5 b(a)30 b Fp(M)2015 2073 y Fl(\032)2085 2058 y Fg(and)g Fu(\()p Fp(M)2413 2073 y Fl(\032)2453 2058 y Fo(;)17 b(\034)2539 2073 y Fl(\032)2579 2058 y Fu(\))31 b Fg(is)g(a)f Fo(W)2935 2022 y Fw(\003)2974 2058 y Fg(-dynamic)-5 b(al)30 b(system.)41 2220 y Fu(\(2\))48 b Fo(L)280 2235 y Fl(\032)356 2220 y Fg(is)35 b(the)g(Liouvil)5 b(le)-5 b(an)33 b(for)i(the)g(dynamics)f Fo(\034)1915 2235 y Fl(\032)1956 2220 y Fg(.)41 2382 y Fu(\(3\))48 b Fg(Assume)35 b(that)h(\(B.46\))d(holds)h(and)h(that) 1343 2515 y Fc(Z)1443 2651 y Fu(\()p Fn(j)p Fo(\030)5 b Fn(j)1585 2610 y Fw(\000)p Fk(1)1700 2651 y Fu(+)22 b Fn(j)p Fo(\030)5 b Fn(j)1902 2610 y Fk(2)1940 2651 y Fu(\))p Fn(k)p Fo(v)t Fu(\()p Fo(\030)g Fu(\))p Fn(k)2253 2610 y Fk(2)2291 2651 y Fu(d)p Fo(\030)33 b(<)27 b Fn(1)p Fo(:)214 2919 y Fg(Then)35 b(\(B.47\))e(holds,)h(and)g(ther)-5 b(e)35 b(exists)g(a)f Fo(\014)6 b Fg(-KMS)35 b(ve)-5 b(ctor)35 b(for)f Fo(\034)2676 2934 y Fl(\032)2717 2919 y Fg(.)146 3147 y Fu(The)d Fo(W)450 3111 y Fw(\003)489 3147 y Fu(-dynamical)g(system)h(\()p Fp(M)1452 3162 y Fl(\032)1492 3147 y Fo(;)17 b(\034)1578 3162 y Fl(\032)1618 3147 y Fu(\))30 b(is)h(called)f(the)h(P)m(auli-Fierz)f(system)i(at)e (densit)m(y)i Fo(\032)p Fu(.)43 b(It)30 b(is)0 3268 y(canonically)k (de\014ned)g(giv)m(en)f Fn(K)q Fu(,)g Fo(K)7 b Fu(,)33 b Fo(v)j Fu(and)d Fo(\032)p Fu(.)146 3388 y(The)46 b(pro)s(of)d(of)h (Theorem)i(B.2)f(is)g(giv)m(en)g(in)g([DJ2)o(].)80 b(T)-8 b(o)44 b(pro)m(v)m(e)i(\(1\))e(w)m(e)h(c)m(hec)m(k)i(that)e Fo(Q)3528 3352 y Fk(semi)3528 3413 y Fl(\032)3714 3388 y Fu(is)0 3509 y(a\016liated)31 b(to)f Fp(M)621 3524 y Fl(\032)692 3509 y Fu(and)h(that)f Fo(L)1155 3472 y Fk(semi)1155 3534 y(fr)1316 3509 y Fu(+)18 b Fo(\025Q)1544 3472 y Fk(semi)1544 3533 y Fl(\032)1716 3509 y Fu(is)32 b(essen)m(tially)h(self-adjoin)m(t)e(on)f Fn(D)s Fu(\()p Fo(L)3095 3472 y Fk(semi)3095 3534 y(fr)3237 3509 y Fu(\))18 b Fn(\\)h(D)s Fu(\()p Fo(Q)3573 3472 y Fk(semi)3573 3533 y Fl(\032)3714 3509 y Fu(\).)0 3629 y(Then)47 b(w)m(e)g(apply)f (Theorem)h(3.3.)82 b(T)-8 b(o)46 b(pro)m(v)m(e)h(\(2\),)i(in)d(a)f (similar)i(w)m(a)m(y)g(w)m(e)f(apply)h(Theorem)g(3.5.)0 3749 y(Finally)-8 b(,)33 b(to)f(sho)m(w)i(\(3\))e(w)m(e)i(use)f (Theorem)h(5.6.)43 b(The)34 b(details)f(can)g(b)s(e)g(found)f(in)h ([DJ2].)146 3870 y(W)-8 b(e)33 b(\014nish)h(with)f(sev)m(eral)h (remarks.)146 3990 y(The)h(p)s(erturbation)e Fo(Q)997 3954 y Fk(semi)997 4015 y Fl(\032)1173 3990 y Fu(is)h(un)m(b)s(ounded)h (from)e(ab)s(o)m(v)m(e)h(and)g(b)s(elo)m(w,)h(and)e(the)h(existing)h (results)0 4110 y(in)e(the)g(literature)g([Ar1,)f(Don,)h(Sa2)o(])g(are) g(not)f(applicable)i(to)e(P)m(auli-Fierz)h(systems.)146 4231 y(The)41 b(\014rst)f(result)g(ab)s(out)g(existence)i(of)d (KMS-states)h(for)f(P)m(auli-Fierz)i(systems)h(go)s(es)d(bac)m(k)i(to)0 4351 y([FNV])i(where)h(the)g(spin-b)s(oson)g(system)h(w)m(as)f (considered.)77 b(A)43 b(result)h(similar)f(to)g(Theorem)i(B.2)0 4472 y(w)m(as)f(pro)m(v)m(en)h(in)f([BFS])f(under)h(a)f(more)h (restrictiv)m(e)i(infrared)d(condition.)77 b(Theorem)44 b(B.2)g(co)m(v)m(ers)0 4592 y(the)37 b(ph)m(ysical)i(infrared)e(regime) g(of)g(non-relativistic)h(QED)e(\(often)h(called)h(the)f(ohmic)g(case)h (in)f(the)0 4712 y(con)m(text)d(of)e(P)m(auli-Fierz)h(systems,)i(see)f (e.g.)44 b([DJ1)o(,)33 b(DJ2)o(,)g(LCD,)f(W)-8 b(e)q(]\).)1841 5753 y(45)p eop %%Page: 46 46 46 45 bop 0 407 a Ft(References)0 626 y Fu([Ar1])133 b(Araki,)38 b(H.:)52 b(Relativ)m(e)38 b(Hamiltonian)g(for)e(a)h (faithful)f(normal)h(states)h(of)e(a)h(v)m(on)g(Neumann)347 746 y(algebra,)32 b(Pub.)i(R.I.M.S.,)f(Ky)m(oto)g(Univ.)g Fj(9)p Fu(,)g(165)f(\(1973\).)0 950 y([Ar2])133 b(Araki,)44 b(H.:)62 b(Golden-Thompson)43 b(and)e(P)m(eierls-Bogoliub)s(o)m(v)j (inequalities)g(for)d(a)g(general)347 1070 y(v)m(on)33 b(Neumann)h(algebra,)e(Comm)m(un.)i(Math.)f(Ph)m(ys.)i Fj(34)p Fu(,)d(167)g(\(1973\).)0 1273 y([Ar3])133 b(Araki,)27 b(H.:)40 b(P)m(ositiv)m(e)26 b(cone,)i(Radon-Nik)m(o)s(dym)d(theorems,) j(relativ)m(e)e(Hamiltonian)f(and)g(the)347 1394 y(Gibbs)35 b(condition)h(in)f(statistical)g(mec)m(hanics.)j(An)d(application)g(of) f(the)i(T)-8 b(omita-T)g(ak)m(esaki)347 1514 y(theory)g(,)32 b(in)g Fo(C)861 1478 y Fw(\003)901 1514 y Fg(-algebr)-5 b(as)33 b(and)g(their)h(Applic)-5 b(ations)34 b(to)g(Statistic)-5 b(al)34 b(Me)-5 b(chanics)33 b(and)h(Quan-)347 1635 y(tum)h(Field)f (The)-5 b(ory,)32 b Fu(\(D.)g(Kastler)h(editor\),)g(Amsterdam,)h (North-Holand)e(\(1976\).)0 1838 y([Ar4])133 b(Araki,)40 b(H.:)56 b(Relativ)m(e)40 b(en)m(trop)m(y)g(of)e(states)h(of)g(v)m(on)g (Neumann)h(algebras,)g(Pub.)f(R.I.M.S.,)347 1958 y(Ky)m(oto)32 b(Univ.)i Fj(11)p Fu(,)f(809)f(\(1976\).)0 2162 y([Ar5])133 b(Araki,)31 b(H.:)42 b(Relativ)m(e)31 b(en)m(trop)m(y)g(of)f(states)h (of)e(v)m(on)i(Neumann)g(algebras)f(I)s(I,)g(Pub.)g(R.I.M.S.,)347 2282 y(Ky)m(oto)i(Univ.)i Fj(13)p Fu(,)f(173)f(\(1977\).)0 2486 y([Ar6])133 b(Araki,)40 b(H.:)56 b(Expansionals)40 b(in)f(Banac)m(h)g(algebras,)h(Ann.)g(Sci.)f(Ecole)g(Norm.)g(Sup.)g Fj(6)p Fu(,)h(67)347 2606 y(\(1973\).)0 2809 y([ArW])82 b(Araki,)31 b(H.,)g(W)-8 b(o)s(o)s(ds,)31 b(E.)f(J.:)43 b(Represen)m(tations)32 b(of)e(the)h(canonical)g(comm)m(utation)g (relations)347 2930 y(describing)j(a)e(nonrelativistic)j(in\014nite)e (free)g(Bose)g(gas,)g(J.)g(Math.)g(Ph)m(ys.)h Fj(4)p Fu(,)f(637)f(\(1963\).)0 3133 y([BaJu])71 b(Baa)5 b(j,)37 b(S.,)h(Jungl,)g(P)-8 b(.:)52 b(Th)m(\023)-46 b(eorie)38 b(biv)-5 b(arian)m(t)36 b(de)h(Kasparo)m(w)g(et)g(op)s(erateurs)g(non)g (b)s(orne'ees)347 3253 y(dans)c(les)g Fo(C)793 3217 y Fw(\003)833 3253 y Fu(-mo)s(dules)g(hilb)s(ertiens,)h(C.)f(R.)g(Acad.)g (Sci.)g(P)m(aris,)h(Serie)f(I)g(296,)f(875)g(\(1983\).)0 3457 y([BFS])106 b(Bac)m(h,)36 b(V.,)f(F)-8 b(r\177)-49 b(ohlic)m(h,)35 b(J.,)h(Sigal,)f(I.:)47 b(Return)35 b(to)f (equilibrium,)j(J.)e(Math.)g(Ph)m(ys.)h Fj(41)p Fu(,)f(3985)347 3577 y(\(2000\).)0 3781 y([BR1])103 b(Brattelli,)39 b(O,)f(Robinson,)h (D.)e(W.:)54 b Fg(Op)-5 b(er)g(ator)39 b(A)n(lgebr)-5 b(as)39 b(and)g(Quantum)g(Statistic)-5 b(al)39 b(Me-)347 3901 y(chanics)34 b(1)p Fu(,)e(Springer-V)-8 b(erlag,)33 b(Berlin,)g(second)h(edition)f(\(1987\).)0 4104 y([BR2])103 b(Brattelli,)39 b(O,)f(Robinson,)h(D.)e(W.:)54 b Fg(Op)-5 b(er)g(ator)39 b(A)n(lgebr)-5 b(as)39 b(and)g(Quantum)g(Statistic)-5 b(al)39 b(Me-)347 4225 y(chanics)34 b(2)p Fu(,)e(Springer-V)-8 b(erlag,)33 b(Berlin,)g(second)h(edition)f(\(1996\).)0 4428 y([Co])174 b(Connes,)44 b(A.:)60 b(Une)42 b(classi\014cation)g (des)g(facteurs)g(de)f(t)m(yp)s(e)h(I)s(I)s(I,)f(Ann.)g(Sci.)h(Ecole)f (Norm.)347 4549 y(Sup.)33 b Fj(6)p Fu(,)g(133)f(\(1973\).)0 4752 y([Don])115 b(Donald,)26 b(M.J.:)41 b(Relativ)m(e)27 b(Hamiltonians)f(whic)m(h)h(are)f(not)g(b)s(ounded)g(from)f(ab)s(o)m(v) m(e,)j(J.)e(F)-8 b(unc.)347 4872 y(Anal.)33 b Fj(91)p Fu(,)f(143)g(\(1990\).)0 5076 y([DJ1])119 b(Derezinski,)29 b(J.,)f(Jak)-5 b(\024)-44 b(s)q(i)m(\023)e(c,)28 b(V.:)41 b(Sp)s(ectral)27 b(theory)g(of)e(P)m(auli-Fierz)j(op)s(erators,)f(J.)g (F)-8 b(unc.)26 b(Anal.)347 5196 y Fj(180)p Fu(,)33 b(241)e(\(2001\).) 1841 5753 y(46)p eop %%Page: 47 47 47 46 bop 0 407 a Fu([DJ2])119 b(Derezinski,)28 b(J.,)e(Jak)-5 b(\024)-44 b(si)m(\023)e(c,)27 b(V.:)39 b(Return)25 b(to)e(equilibrium) k(for)c(P)m(auli-Fierz)i(systems,)j(preprin)m(t,)347 527 y(Maph)m(ysto,)34 b(Aarh)m(us)f(Univ)m(ersit)m(y)j(\(2001\).)0 731 y([FNV])83 b(F)-8 b(annes,)42 b(M.,)g(Nac)m(h)m(tergale,)g(B.,)g(V) -8 b(erb)s(eure,)42 b(A.:)58 b(The)40 b(equilibrium)i(states)e(of)f (the)h(spin-)347 851 y(b)s(oson)32 b(mo)s(del,)i(Comm.)f(Math.)g(Ph)m (ys.)h Fj(114)p Fu(,)f(537)f(\(1988\).)0 1054 y([Haa])122 b(Haagerup,)39 b(U.:)55 b(The)39 b(standard)f(form)g(of)g(v)m(on)g (Neumann)h(algebras,)h(Math.)f(Scand.)f Fj(37)p Fu(,)347 1175 y(271)32 b(\(1975\).)0 1378 y([JP1])128 b(Jak)-5 b(\024)-44 b(s)q(i)m(\023)e(c,)33 b(V.,)g(Pillet)g(C.-A.:)44 b(On)32 b(a)h(mo)s(del)g(for)f(quan)m(tum)i(friction)e(I)s(I)s(I.)h (Ergo)s(dic)g(prop)s(erties)347 1499 y(of)f(the)h(spin-b)s(oson)g (system,)i(Comm)m(un.)f(Math.)f(Ph)m(ys.)h Fj(178)p Fu(,)f(627)f (\(1996\).)0 1702 y([JP2])128 b(Jak)-5 b(\024)-44 b(s)q(i)m(\023)e(c,) 25 b(V.,)g(Pillet,)h(C.-A.:)39 b(Non-equilibrium)25 b(steady)f(states)g (for)e(\014nite)i(quan)m(tum)g(systems)347 1822 y(coupled)34 b(to)e(thermal)h(reserv)m(oirs,)i(to)d(app)s(ear)g(in)h(Comm)m(un.)h (Math.)f(Ph)m(ys.)0 2026 y([KL])156 b(Klein,)31 b(A.,)f(Landau,)g(L.:) 42 b(Sto)s(c)m(hastic)31 b(pro)s(cesses)h(asso)s(ciated)e(with)g(KMS)g (states,)h(J.)f(F)-8 b(unc.)347 2146 y(Anal.)33 b Fj(42)p Fu(,)f(368)g(\(1981\).)0 2350 y([LCD])87 b(Legget,)40 b(A.J.,)g(Chakra)m(v)-5 b(art)m(y)d(,)42 b(S.,)e(Dorsey)-8 b(,)40 b(A.T.,)h(Fisher)e(M.P)-8 b(.A.,)41 b(Garg,)e(A.,)i(Zw)m(erger,) 347 2470 y(W.:)g(Dynamics)30 b(of)e(the)h(dissipativ)m(e)h(t)m(w)m (o-state)f(system,)j(Rev.)d(Mo)s(d.)f(Ph)m(ys.)i Fj(59)p Fu(,)f(1)f(\(1987\).)0 2673 y([M])204 b(Merkli,)50 b(M.:)69 b(P)m(ositiv)m(e)48 b(comm)m(utators)e(in)g(non-equilibrium)h(quan)m (tum)f(statistical)g(me-)347 2794 y(c)m(hanics,)34 b(preprin)m(t.)0 2997 y([OP])151 b(Oh)m(y)m(a,)57 b(M.,)g(P)m(etz,)h(D.:)81 b Fg(Quantum)53 b(Entr)-5 b(opy)52 b(and)g(its)h(Use)p Fu(,)j(Springer-V)-8 b(erlag,)57 b(Berlin)347 3117 y(\(1993\).)0 3321 y([PuW)-8 b(o])49 b(Pusz,)36 b(W.,)e(W)-8 b(orono)m(wicz,)35 b(S.)f(L.:)46 b(F)-8 b(orm)33 b(con)m(v)m(ex)j(functions)e(and)g(the)g (WYDL)g(and)f(other)347 3441 y(inequalities,)i(Lett.)d(Math.)h(Ph)m (ys.)i Fj(2)p Fu(,)d(505)g(\(1978\).)0 3645 y([RS1])118 b(Reed,)47 b(M.,)g(Simon,)h(B.:)66 b Fg(Metho)-5 b(ds)45 b(of)g(Mo)-5 b(dern)45 b(Mathematic)-5 b(al)44 b(Physics)h(I.)g(F)-7 b(unctional)347 3765 y(A)n(nalysis)p Fu(,)32 b(Academic)i(Press,)g(San) f(Diego)f(\(1973\).)0 3968 y([RS2])118 b(Reed,)29 b(M.,)g(Simon,)h(B.:) 41 b Fg(Metho)-5 b(ds)30 b(of)g(Mo)-5 b(dern)30 b(Mathematic)-5 b(al)30 b(Physics)g(II.)g(F)-7 b(ourier)29 b(A)n(nal-)347 4089 y(ysis,)35 b(Self-A)-5 b(djointness)p Fu(,)31 b(Academic)j(Press,) g(San)f(Diego)f(\(1975\).)0 4292 y([RZ])161 b(Raggio,)41 b(G.A.,)h(Zivi,)g(S.H.:)60 b(Semiclassical)42 b(description)g(of)e Fo(N)10 b Fu(-lev)m(el)41 b(systems)i(in)m(teract-)347 4413 y(ing)33 b(with)g(radiation)f(\014elds,)j(in)e("Quan)m(tum)g(P)m (orbabilit)m(y)h(I)s(I,)f(Heidelb)s(erg)h(1984",)d(Lecture)347 4533 y(Notes)i(in)g(Math.,)g(V)-8 b(ol.)32 b(1136,)g(Springer-V)-8 b(erlag,)33 b(Berlin-New)h(Y)-8 b(ork,)32 b(1985.)0 4736 y([Sa1])141 b(Sak)-5 b(ai,)42 b(S.:)59 b Fg(Op)-5 b(er)g(ator)41 b(A)n(lgebr)-5 b(as)41 b(in)h(Dynamic)-5 b(al)40 b(Systems.)i(The)f (The)-5 b(ory)42 b(of)f(Unb)-5 b(ounde)g(d)347 4857 y(Derivations)44 b(in)g Fo(C)1089 4821 y Fw(\003)1129 4857 y Fg(-algebr)-5 b(as.)42 b Fu(Encyclop)s(edia)k(of)d(Mathematics)i(and)f(its)g (Applications)347 4977 y Fj(41)p Fu(,)33 b(Cam)m(bridge)g(Univ)m(ersit) m(y)j(Press,)e(Cam)m(bridge)g(\(1991\).)1841 5753 y(47)p eop %%Page: 48 48 48 47 bop 0 407 a Fu([Sa2])141 b(Sak)-5 b(ai,)36 b(S:)g(P)m (erturbations)h(of)e(KMS)g(states)i(in)f Fo(C)2215 371 y Fw(\003)2254 407 y Fu(-dynamical)g(systems)i(\(Generalization)347 527 y(of)26 b(the)h(absence)h(theorem)g(of)e(phase)i(transition)f(to)f (con)m(tin)m(uous)i(quan)m(tum)g(systems\),)j(Con-)347 648 y(temp.)i(Math.)g Fj(62)p Fu(,)g(187)f(\(1987\).)0 851 y([Sa3])141 b(Sak)-5 b(ai,)32 b(S:)h Fo(C)822 815 y Fw(\003)862 851 y Fg(-algebr)-5 b(as)33 b(and)i Fo(W)1562 815 y Fw(\003)1601 851 y Fg(-algebr)-5 b(as)p Fu(,)31 b(Springer-V)-8 b(erlag,)33 b(Berlin)g(\(1971\).)0 1054 y([Si])212 b(Simon,)33 b(B.:)44 b Fg(Statistic)-5 b(al)35 b(Me)-5 b(chanics)34 b(of)h(L)-5 b(attic)g(e)35 b(Gasses,)g(V)-7 b(olume)34 b(1)p Fu(,)f(Princeton)h(Univ)m(er-)347 1175 y(sit)m(y)g(Press,)g(Princeton)g(\(1991\).)0 1378 y([St])201 b(Stratila,)29 b(S.:)41 b Fg(Mo)-5 b(dular)31 b(the)-5 b(ory)31 b(in)g(op)-5 b(er)g(ator)31 b(algebr)-5 b(as)p Fu(,)28 b(Abacus)h(Press,)i(T)-8 b(urn)m(bridge)29 b(W)-8 b(ells)347 1499 y(\(1981\).)0 1702 y([StZs])103 b(Stratila,)37 b(S.,)h(Zsido,)g(L.:)52 b Fg(L)-5 b(e)g(ctur)g(es)39 b(on)f(von)g(Neumann)h(algebr)-5 b(as)p Fu(,)36 b(Abacus)i(Press,)h(T) -8 b(urn-)347 1822 y(bridge)33 b(W)-8 b(ells)34 b(\(1979\).)0 2026 y([Uh])166 b(Uhlmann,)34 b(A.:)46 b(Relativ)m(e)34 b(en)m(trop)m(y)h(and)f(the)g(Wigner-Y)-8 b(anase-Dyson-Lieb)34 b(conca)m(vit)m(y)h(in)347 2146 y(an)d(in)m(terp)s(olation)h(theory)-8 b(,)33 b(Comm)m(un.)h(Math.)f(Ph)m(ys.)i Fj(54)p Fu(,)e(123)e (\(1977\).)0 2350 y([W)-8 b(e])158 b(W)-8 b(eiss,)46 b(U.:)62 b Fg(Quantum)43 b(Dissip)-5 b(ative)42 b(Systems)p Fu(,)i(Series)g(in)e(Mo)s(dern)g(Condensed)i(Matter)347 2470 y(Ph)m(ysics-V)-8 b(ol.)35 b(2,)d(W)-8 b(orld)33 b(Scien)m(ti\014c,)h(Singap)s(ore,)f(second)h(enlarged)f(edition)g (1999.)0 2673 y([W)-8 b(o])152 b(W)-8 b(orono)m(wicz,)50 b(S.)c(L.:)70 b(Un)m(b)s(ounded)47 b(elemen)m(ts)h(a\017liated)f(with)f Fo(C)2922 2637 y Fw(\003)2961 2673 y Fu(-algebras)g(and)g(non-)347 2794 y(compact)33 b(quan)m(tum)h(groups,)f(Comm)m(un.)h(Math.)f(Ph)m (ys.)h Fj(136)p Fu(,)f(399)f(\(1991\).)1841 5753 y(48)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF