Content-Type: multipart/mixed; boundary="-------------0308140720831" This is a multi-part message in MIME format. ---------------0308140720831 Content-Type: text/plain; name="03-367.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-367.keywords" pairing of K-theory with cyclic cohomology, quantum Hall effect ---------------0308140720831 Content-Type: application/postscript; name="topo4.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="topo4.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.92b Copyright 2002 Radical Eye Software %%Title: topo4.dvi %%Pages: 34 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMR17 CMMI12 CMSY10 MSBM10 CMR12 CMR10 CMBX10 CMMI10 %%+ CMBX12 CMSY8 CMTI12 CMMI8 CMR8 CMSY6 CMEX10 CMSL12 CMR7 CMR6 MSBM7 %%+ CMMI6 LASY10 CMR5 CMSY7 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips topo4 %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2003.07.24:0940 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All Rights Reserved) readonly def /FullName (CMSY7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY7 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 /.notdef put readonly def /FontBBox{-15 -951 1252 782}readonly def /UniqueID 5000817 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR5 %!PS-AdobeFont-1.1: CMR5 1.00B %%CreationDate: 1992 Feb 19 19:55:02 % 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 (CMR5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR5 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 /.notdef put readonly def /FontBBox{-341 -250 1304 965}readonly def /UniqueID 5000788 def currentdict end currentfile eexec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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 0 /.notdef put readonly def /FontBBox{-32 -250 1048 750}readonly def /UniqueID 5087385 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: LASY10 %!PS-AdobeFont-1.1: LASY10 1.001 %%CreationDate: 1992 Oct 23 20:19:17 %%RevisionDate: 2001 Jun 05 20:19:17 % Copyright (C) 1997, 2001 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.001) readonly def /Notice (Copyright (C) 1997, 2001 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 /UniqueID 5011949 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: MSBM7 %!PS-AdobeFont-1.1: MSBM7 2.1 %%CreationDate: 1992 Oct 17 08:30:50 % 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 (MSBM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM7 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 78 /N put dup 82 /R put dup 126 /planckover2pi1 put readonly def /FontBBox{0 -504 2615 1004}readonly def /UniqueID 5032014 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF5B8CABB9FFC6A66A4000A13D5F68BFF326D 1D432B0D064B56C598F4338C319309181D78E1629A31ECA5DD8536379B03C383 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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. 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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 72 /H put dup 82 /R put dup 111 /multicloseright put dup 126 /planckover2pi1 put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueID 5031982 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF5B8CABB9FFC6A66A4000A13D5F68BFF326D 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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 (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 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 /.notdef put readonly def /FontBBox{11 -250 1241 750}readonly def /UniqueID 5087381 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D6A8F05B47AF95EF28A9C561DBDC98C47CF5 5250011D19E9366EB6FD153D3A100CAA6212E3D5D93990737F8D326D347B7EDC 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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 0 /.notdef put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueID 5000789 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF4E9D2405B169CD5365D6ECED5D768D66D6C 68618B8C482B341F8CA38E9BB9BAFCFAAD9C2F3FD033B62690986ED43D9C9361 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR7 %!PS-AdobeFont-1.1: CMR7 1.0 %%CreationDate: 1991 Aug 20 16:39:21 % 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 (CMR7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR7 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 /.notdef put readonly def /FontBBox{-27 -250 1122 750}readonly def /UniqueID 5000790 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF5B8CABB9FFC6CC3F1E9AE32F234EB60FE7D E34995B1ACFF52428EA20C8ED4FD73E3935CEBD40E0EAD70C0887A451E1B1AC8 47AEDE4191CCDB8B61345FD070FD30C4F375D8418DDD454729A251B3F61DAE7C 8882384282FDD6102AE8EEFEDE6447576AFA181F27A48216A9CAD730561469E4 78B286F22328F2AE84EF183DE4119C402771A249AAC1FA5435690A28D1B47486 1060C8000D3FE1BF45133CF847A24B4F8464A63CEA01EC84AA22FD005E74847E 01426B6890951A7DD1F50A5F3285E1F958F11FC7F00EE26FEE7C63998EA1328B C9841C57C80946D2C2FC81346249A664ECFB08A2CE075036CEA7359FCA1E90C0 F686C3BB27EEFA45D548F7BD074CE60E626A4F83C69FE93A5324133A78362F30 8E8DCC80DD0C49E137CDC9AC08BAE39282E26A7A4D8C159B95F227BDA2A281AF A9DAEBF31F504380B20812A211CF9FEB112EC29A3FB3BD3E81809FC6293487A7 455EB3B879D2B4BD46942BB1243896264722CB59146C3F65BD59B96A74B12BB2 9A1354AF174932210C6E19FE584B1B14C00E746089CBB17E68845D7B3EA05105 EEE461E3697FCF835CBE6D46C75523478E766832751CF6D96EC338BDAD57D53B 52F5340FAC9FE0456AD13101824234B262AC0CABA43B62EBDA39795BAE6CFE97 563A50AAE1F195888739F2676086A9811E5C9A4A7E0BF34F3E25568930ADF80F 0BDDAC3B634AD4BA6A59720EA4749236CF0F79ABA4716C340F98517F6F06D9AB 7ED8F46FC1868B5F3D3678DF71AA772CF1F7DD222C6BF19D8EF0CFB7A76FC6D1 0AD323C176134907AB375F20CFCD667AB094E2C7CB2179C4283329C9E435E7A4 1E042AD0BAA059B3F862236180B34D3FCED833472577BACD472A4A78141CA32C B3C74E1A0AE0520B950B826B0ABD81766035058ED1654D00FE541CAD1D246C0E DE85FCD3C0BF7A70B913487B1A527EA823C00C39DB61FD6641B140FCED8580D8 046741D2494B4E7CA1F120CBB0A532BE049CABEB70A39018E8212F8178E93C98 B377AE2880FE39BA0EE29451857DB34964DA26ACA4CD23CE284ACE37D89571FF CB67DE7AE379B74B32ECACC5F3DE0566CE9EE820E96F27653C75935851CD9360 A83C7EE8270383CB8A80715BC2B62B1F709235A5A73D99710D7665182D461095 B98C8A0FEA44F0F1959055D50BABC7880E7BA1CA4CD72531A240A622663A0A1F DBE4FB907F97515CB1100282C9A0241F65F84EBAB1701FC105BD25F82807F4CB 6A5EDB5CB156A7D55F64146818245C112DB0FEE9E0AC96B4B2AEB27F89FE0560 727D85FE6CFF5C457EB1EE5D7E2A09979684E2611BA57A1BC7BA4E37DC4BA761 557D986F9A8B495CB7212507AA79C297B0665CB5883F2332DF5CB088A92E24BD 1EBADCAA515B567FAA9E15E7B8DA60C1BBEEF5A2E0D4C0C67EDAD822B5E8D81C 6D29928726D36EF0A9238476FE54D4990E8D75FE0109C0336DD50F9021307213 F69C700D2291C546007CF3A5DF3BF5ABEEB640D3AEF585055A65EB2CA7AD7F11 22182523EAA8FCF029C604212529C03F179566E6B731FFE2283D6402A350EAB9 3C7FFF41383A9398B33C48FB2F9A8241D65425FC5E49DC9BB97521D91E44B3C8 B2EB1BA8D532BBF175486D1DDB914BCB92968D342B5C1FD6FC72DA462FF68992 464BF0E00D05C2AC3729E598991C6EE7354F0C400296356ABAB37A16FA504DC9 B6369F88BDB7061D5EEB761527E588B4A6C83169B517991789D664BB543A0E57 6F2529BDC7838AFB93D53D5794E57B65A7DB00584492C046F72CE19E4D1B177D 1D951362B2F7CE2B720B305D45B7FC74183C1DE46073D8E20FDF8027FBFCDF40 5F0950F5AAC19AC8E8D9A57A271C0BB212822798DCD068B4F05DBAC2AF7BB25F 5DFBFBFB5A51ED26ACB22541E1971002A5D97B5E148F8F087A3229C4 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 0 /.notdef put readonly def /FontBBox{-53 -251 1139 750}readonly def /UniqueID 5000769 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5F0364CD5660F74BEE96790DE35AFA90CCF712 B1805DA88AE375A04D99598EADFC625BDC1F9C315B6CF28C9BD427F32C745C99 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % 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 (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 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 20 /bracketleftbigg put dup 21 /bracketrightbigg put dup 48 /parenlefttp put dup 49 /parenrighttp put dup 56 /bracelefttp put dup 58 /braceleftbt put dup 60 /braceleftmid put dup 64 /parenleftbt put dup 65 /parenrightbt put dup 76 /circleplustext put dup 77 /circleplusdisplay put dup 80 /summationtext put dup 82 /integraltext put dup 83 /uniontext put dup 84 /intersectiontext put dup 88 /summationdisplay put dup 90 /integraldisplay put dup 91 /uniondisplay put dup 92 /intersectiondisplay put dup 122 /bracehtipdownleft put dup 123 /bracehtipdownright put dup 124 /bracehtipupleft put dup 125 /bracehtipupright put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueID 5000774 def currentdict end currentfile eexec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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 /.notdef put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueID 5000816 def currentdict end currentfile eexec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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 0 /.notdef put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueID 5000791 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF4E9D2405B169CD5365D6ECED5D768D66D6C 68618B8C482B341F8CA38E9BB9BAFCFAAD9C2F3FD033B62690986ED43D9C9361 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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. 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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 /.notdef put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueID 5000818 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4 A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85 E21B3ABF72E39A89232CD9F4237FC80C9E64E8425AA3BEF7DED60B122A52922A 221A37D9A807DD01161779DDE7D5FC1B2109839E5B52DFBB2A7C1B5D8E7E8AA0 5B10EA43D6A8ED61AF5B23D49920D8F79DAB6A59062134D84AC0100187A6CD1F 80F5DDD9D222ACB1C23326A7656A635C4A241CCD32CBFDF8363206B8AA36E107 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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. 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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 0 /.notdef put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueID 5000768 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5F00F963068B8B731A88D7740B0DDAED1B3F82 7DB9DFB4372D3935C286E39EE7AC9FB6A9B5CE4D2FAE1BC0E55AE02BFC464378 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(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 0 /.notdef put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueID 5000793 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF7158F1163BC1F3352E22A1452E73FECA8A4 87100FB1FFC4C8AF409B2067537220E605DA0852CA49839E1386AF9D7A1A455F 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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. 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Fz(-algebra)e Ft(B)s Fz(,)i Fy(\013)3297 5013 y Fk(~)3368 5049 y Fz(:)30 b Fu(R)f Ft(!)g Fz(Aut)q(\()p Ft(B)s Fz(\).)-24 5169 y(Collecting)34 b(these)g(together)f(w)m(e)g (get)g(an)f Fu(R)h Fz(action)42 b(~)-59 b Fy(\013)29 b Fz(:)f Fu(R)f Ft(!)g Fz(Aut)q(\()p Fy(C)2556 5184 y Fr(0)2595 5169 y Fz(\()p Fy(I)8 b(;)17 b Ft(B)s Fz(\)\))32 b(b)m(y)1538 5362 y(~)-58 b Fy(\013)1591 5377 y Fs(t)1620 5362 y Fz(\()p Fy(f)11 b Fz(\)\()p Fu(~)p Fz(\))27 b(=)h Fy(\013)2079 5321 y Fk(~)2078 5387 y Fs(t)2121 5362 y Fz(\()p Fy(f)11 b Fz(\()p Fu(~)p Fz(\)\))27 b Fy(;)1935 5611 y Fz(11)p eop end %%Page: 12 12 TeXDict begin 12 11 bop -24 171 a Fz(whic)m(h)36 b(is)f(con)m(tin)m (uous)g(pro)m(vided)h(the)e(ab)s(o)m(v)m(e)h(expression)h(is)f(con)m (tin)m(uous)h(in)e Fu(~)g Fz(for)f(all)h Fy(t)h Fz(and)f Fy(f)44 b Fz(whic)m(h)36 b(w)m(e)-24 291 y(hereb)m(y)41 b(assume.)64 b(Then)40 b(\()p Fy(C)1060 306 y Fr(0)1099 291 y Fz(\()p Fy(I)8 b(;)17 b Ft(B)s Fz(\))26 b Fu(o)1448 306 y Fr(~)-42 b Fs(\013)1518 291 y Fu(R)p Fy(;)17 b Ft(fB)29 b Fu(o)1855 310 y Fs(\013)1900 291 y Fe(~)36 b Fu(R)p Fy(;)17 b Fz(ev)2180 306 y Fk(~)2223 291 y Ft(g)2273 306 y Fk(~)p Fx(2)p Fs(I)2398 291 y Fz(\))38 b(is)i(a)e(con)m(tin)m (uous)j(\014eld)e(of)g Fy(C)3573 255 y Fx(\003)3612 291 y Fz(-algebras)-24 411 y([R89].)-24 640 y Fw(Example)g(4)e(\(Heisen)m (b)s(erg)h(group)g(algebra\))49 b Fz(The)i(\(p)s(olarized\))f(Heisen)m (b)s(erg)i(group)e Fu(H)3539 655 y Fr(3)3628 640 y Fz(is)h Fu(R)3816 604 y Fr(3)3905 640 y Fz(as)-24 760 y(top)s(ological)32 b(space,)i(but)f(with)g(\(non-ab)s(elian\))f(m)m(ultiplication)806 980 y(\()p Fy(a)895 995 y Fr(1)934 980 y Fy(;)17 b(a)1029 995 y Fr(2)1069 980 y Fy(;)g(a)1164 995 y Fr(3)1203 980 y Fz(\)\()p Fy(b)1320 995 y Fr(1)1360 980 y Fy(;)g(b)1445 995 y Fr(2)1484 980 y Fy(;)g(b)1569 995 y Fr(3)1609 980 y Fz(\))27 b(=)h(\()p Fy(a)1867 995 y Fr(1)1929 980 y Fz(+)22 b Fy(b)2068 995 y Fr(1)2108 980 y Fy(;)17 b(a)2203 995 y Fr(2)2264 980 y Fz(+)22 b Fy(b)2403 995 y Fr(2)2443 980 y Fy(;)17 b(a)2538 995 y Fr(3)2600 980 y Fz(+)22 b Fy(b)2739 995 y Fr(3)2801 980 y Fz(+)g Fy(a)2950 995 y Fr(1)2989 980 y Fy(b)3030 995 y Fr(2)3070 980 y Fz(\))28 b Fy(:)-24 1200 y Fz(It)34 b(con)m(tains)g(the)g(subgroup)f Fu(R)1130 1164 y Fr(2)1199 1200 y Fz(=)28 b Ft(f)p Fz(\()p Fy(a)1442 1215 y Fr(1)1482 1200 y Fy(;)17 b(a)1577 1215 y Fr(2)1616 1200 y Fy(;)g(a)1711 1215 y Fr(3)1750 1200 y Fz(\))29 b Ft(2)g Fu(H)1989 1215 y Fr(3)2029 1200 y Ft(j)p Fy(a)2108 1215 y Fr(1)2176 1200 y Fz(=)g(0)p Ft(g)j Fz(so)i(that)f Fu(H)2822 1215 y Fr(3)2895 1200 y Fz(can)g(b)s(e)h(iden) m(ti\014ed)h(with)f(the)-24 1320 y(semi-direct)f(pro)s(duct)f Fu(R)911 1284 y Fr(2)970 1320 y Fu(o)1051 1335 y Fr(~)-39 b Fs(\034)1111 1320 y Fu(R)31 b Fz(where)38 b(~)-54 b Fy(\034)1537 1335 y Fs(a)1574 1344 y Fl(1)1613 1320 y Fz(\()p Fy(a)1702 1335 y Fr(2)1742 1320 y Fy(;)17 b(a)1837 1335 y Fr(3)1876 1320 y Fz(\))28 b(=)f(\()p Fy(a)2134 1335 y Fr(2)2174 1320 y Fy(;)17 b(a)2269 1335 y Fr(3)2328 1320 y Fz(+)j Fy(a)2475 1335 y Fr(1)2515 1320 y Fy(a)2566 1335 y Fr(2)2605 1320 y Fz(\).)43 b(The)33 b(Heisen)m(b)s(erg)g(group)e (algebra)-24 1441 y(\(i.e.)h(the)g(crossed)h(pro)s(duct)f Fu(C)20 b(o)1207 1456 y Fl(id)1282 1441 y Fu(H)1359 1456 y Fr(3)1431 1441 y Fz(de\014ned)32 b(in)g(a)f(similar)h(w)m(a)m(y)h(as) f(for)f Fu(R)p Fz(\))g(can)h(therefore)g(b)s(e)f(iden)m(ti\014ed)-24 1561 y(with)c(the)f Fy(C)430 1525 y Fx(\003)469 1561 y Fz(-algebra)f Fy(C)906 1576 y Fr(0)945 1561 y Fz(\()p Fu(R)1055 1525 y Fr(2)1095 1561 y Fz(\))8 b Fu(o)1222 1576 y Fr(~)-39 b Fs(\034)1269 1561 y Fu(R)26 b Fz(with)31 b(~)-54 b Fy(\034)1624 1576 y Fs(a)1661 1585 y Fl(1)1700 1561 y Fz(\()p Fy(f)11 b Fz(\)\()p Fy(a)1924 1576 y Fr(2)1964 1561 y Fy(;)17 b(a)2059 1576 y Fr(3)2098 1561 y Fz(\))28 b(=)f Fy(f)11 b Fz(\()p Fy(a)2415 1576 y Fr(2)2454 1561 y Fy(;)17 b(a)2549 1576 y Fr(3)2597 1561 y Ft(\000)8 b Fy(a)2733 1576 y Fr(1)2773 1561 y Fy(a)2824 1576 y Fr(2)2863 1561 y Fz(\).)41 b(Let)26 b Fy(')3201 1576 y Fs(a)3238 1585 y Fl(2)3305 1561 y Fz(:)i Fy(C)3430 1576 y Fr(0)3469 1561 y Fz(\()p Fu(R)3579 1525 y Fr(2)3618 1561 y Fz(\))8 b Fu(o)3745 1576 y Fr(~)-39 b Fs(\034)3793 1561 y Fu(R)27 b Ft(!)-24 1682 y Fy(C)46 1697 y Fr(0)86 1682 y Fz(\()p Fu(R)p Fz(\))17 b Fu(o)328 1697 y Fs(\034)367 1674 y Fj(a)401 1689 y Fl(2)462 1682 y Fu(R)30 b Fz(b)s(e)h(ev)-5 b(aluation)31 b(of)f(the)h(2-comp)s(onen)m(t)f(at)g Fy(a)2182 1697 y Fr(2)2222 1682 y Fz(,)h(i.e.)g Fy(')2499 1697 y Fs(a)2536 1706 y Fl(2)2575 1682 y Fz(\()p Fy(f)11 b Fz(\)\()p Fy(a)2799 1697 y Fr(1)2838 1682 y Fz(\)\()p Fy(a)2965 1697 y Fr(3)3005 1682 y Fz(\))28 b(=)f Fy(f)11 b Fz(\()p Fy(a)3322 1697 y Fr(1)3361 1682 y Fz(\)\()p Fy(a)3488 1697 y Fr(2)3528 1682 y Fy(;)17 b(a)3623 1697 y Fr(3)3662 1682 y Fz(\).)43 b(Then)-24 1802 y(im)17 b(\()p Fy(')203 1817 y Fs(a)240 1826 y Fl(2)279 1802 y Fz(\))358 1774 y Ft(\030)359 1806 y Fz(=)477 1802 y Fy(C)547 1817 y Fr(0)586 1802 y Fz(\()p Fu(R)p Fz(\))28 b Fu(o)839 1817 y Fs(\034)878 1795 y Fj(a)912 1810 y Fl(2)982 1802 y Fu(R)41 b Fz(where)h Fy(\034)1438 1766 y Fs(a)1475 1775 y Fl(2)1427 1827 y Fs(a)1464 1836 y Fl(1)1514 1802 y Fz(\()p Fy(g)t Fz(\)\()p Fy(a)1730 1817 y Fr(3)1769 1802 y Fz(\))f(=)h Fy(g)t Fz(\()p Fy(a)2106 1817 y Fr(3)2172 1802 y Ft(\000)28 b Fy(a)2328 1817 y Fr(2)2368 1802 y Fy(a)2419 1817 y Fr(1)2458 1802 y Fz(\))41 b(for)f Fy(g)45 b Fz(:)c Fu(R)g Ft(!)g Fy(C)3178 1817 y Fr(0)3217 1802 y Fz(\()p Fu(R)p Fz(\).)68 b(F)-8 b(urthermore)-24 1922 y(\()p Fy(C)84 1937 y Fr(0)124 1922 y Fz(\()p Fu(R)234 1886 y Fr(2)273 1922 y Fz(\))14 b Fu(o)406 1937 y Fr(~)-39 b Fs(\034)459 1922 y Fu(R)p Fy(;)17 b Ft(f)p Fy(C)695 1937 y Fr(0)734 1922 y Fz(\()p Fu(R)p Fz(\))d Fu(o)973 1937 y Fs(\034)1012 1915 y Fj(a)1046 1930 y Fl(2)1103 1922 y Fu(R)p Fy(;)j(')1283 1937 y Fs(a)1320 1946 y Fl(2)1359 1922 y Ft(g)1409 1937 y Fs(a)1446 1946 y Fl(2)1481 1937 y Fx(2)p Fk(R)1580 1922 y Fz(\))28 b(is)h(a)g Fy(C)1895 1886 y Fx(\003)1934 1922 y Fz(-\014eld.)43 b(Therefore)29 b Fy(a)2703 1937 y Fr(2)2772 1922 y Fz(pla)m(ys)g(the)g(role)g(of)f Fu(~)p Fz(.)42 b(Note)29 b(also)-24 2043 y(that)j Fy(C)257 2058 y Fr(0)297 2043 y Fz(\()p Fu(R)p Fz(\))21 b Fu(o)543 2058 y Fs(\034)582 2036 y Fj(a)616 2051 y Fl(2)680 2043 y Fu(R)780 2015 y Ft(\030)781 2047 y Fz(=)885 2043 y Fy(C)962 2007 y Fx(\003)955 2067 y Fs(\015)1002 2043 y Fz(\()p Fy(D)1121 2058 y Fr(1)1160 2043 y Fy(;)c(D)1285 2058 y Fr(2)1324 2043 y Fz(\))32 b(if)g Fy(\015)h Fz(=)28 b Fy(a)1722 2058 y Fr(2)1793 2043 y Fz(\(cf.)33 b(Example)h(2\).)43 b(In)32 b(this)h(situation,)g Fy(a)3322 2058 y Fr(1)3362 2043 y Fy(;)17 b(a)3457 2058 y Fr(3)3528 2043 y Fz(corresp)s(ond)-24 2163 y(to)27 b(the)g(space)h(directions)g(of)f Fy(D)1133 2178 y Fr(1)1172 2163 y Fy(;)17 b(D)1297 2178 y Fr(2)1363 2163 y Fz(and)27 b(the)g Fy(C)1786 2127 y Fx(\003)1826 2163 y Fz(-\014eld)g Ft(F)36 b Fz(of)27 b(Section)g(2.2)g(is)g(the)h (restriction)g(of)e(the)h(ab)s(o)m(v)m(e)-24 2283 y Fy(C)53 2247 y Fx(\003)93 2283 y Fz(-\014eld)33 b(to)f Fy(a)507 2298 y Fr(2)574 2283 y Fz(=)c Fy(\015)k Ft(2)d Fz([0)p Fy(;)17 b Fz(1].)-24 2512 y Fw(Example)39 b(5)49 b Fz(If)32 b(w)m(e)i(ha)m(v)m(e)h(an)d Fu(R)p Fz(-action)h Fy(\013)g Fz(on)g(a)g Fy(C)1934 2476 y Fx(\003)1973 2512 y Fz(-algebra)f Ft(B)k Fz(w)m(e)e(can)f(extend)i(the)e(ab)s(o)m(v)m(e)h(\014eld)f(of)g (the)-24 2632 y(Heisen)m(b)s(erg)40 b(group)e(algebra)f(in)h(the)h (follo)m(wing)f(w)m(a)m(y:)55 b(With)38 b(the)h(ab)s(o)m(v)m(e)f Fu(R)p Fz(-action)43 b(~)-54 b Fy(\034)49 b Fz(on)38 b Fy(C)3479 2647 y Fr(0)3518 2632 y Fz(\()p Fu(R)3628 2596 y Fr(2)3667 2632 y Fz(\))g(de\014ne)-19 2753 y(~)-54 b Fy(\034)40 b Ft(\012)28 b Fy(\013)42 b Fz(:)g Fu(R)f Ft(!)h Fz(Aut)17 b Fy(C)843 2768 y Fr(0)882 2753 y Fz(\()p Fu(R)992 2716 y Fr(2)1031 2753 y Fy(;)g Ft(B)s Fz(\))41 b(b)m(y)h(\()5 b(~)-54 b Fy(\034)39 b Ft(\012)28 b Fy(\013)q Fz(\))1691 2768 y Fs(a)1728 2777 y Fl(1)1767 2753 y Fz(\()p Fy(f)11 b Fz(\)\()p Fy(a)1991 2768 y Fr(2)2030 2753 y Fy(;)17 b(a)2125 2768 y Fr(3)2164 2753 y Fz(\))42 b(=)f Fy(\013)2423 2768 y Fs(a)2460 2777 y Fl(1)2500 2753 y Fz(\()p Fy(f)11 b Fz(\()p Fy(a)2686 2768 y Fr(2)2725 2753 y Fy(;)17 b(a)2820 2768 y Fr(3)2887 2753 y Ft(\000)28 b Fy(a)3043 2768 y Fr(2)3083 2753 y Fy(a)3134 2768 y Fr(1)3173 2753 y Fz(\)\).)68 b(Setting)42 b Fy(a)3740 2768 y Fr(2)3821 2753 y Fz(=)f Fu(~)-24 2873 y Fz(as)33 b(ab)s(o)m(v)m(e,)h(this)g(then)f(yields)i(a)e Fy(C)1247 2837 y Fx(\003)1286 2873 y Fz(-\014eld)g(\()p Fy(C)1638 2888 y Fr(0)1677 2873 y Fz(\()p Fu(R)1787 2837 y Fr(2)1827 2873 y Fy(;)17 b Ft(B)s Fz(\))22 b Fu(o)2080 2888 y Fr(~)-39 b Fs(\034)8 b Fx(\012)p Fs(\013)2242 2873 y Fu(R)p Fy(;)17 b Ft(f)p Fy(C)2478 2888 y Fr(0)2517 2873 y Fz(\()p Fu(R)p Fy(;)g Ft(B)s Fz(\))22 b Fu(o)2876 2892 y Fs(\034)2915 2873 y Fe(~)2954 2892 y Fx(\012)p Fs(\013)3081 2873 y Fu(R)p Fy(;)17 b(')3261 2888 y Fk(~)3303 2873 y Ft(g)3353 2888 y Fk(~)p Fx(2)p Fk(R)3490 2873 y Fz(\))33 b(whic)m(h)h(will)-24 2993 y(b)s(e)42 b(of)f(crucial)h(imp)s(ortance)g(later)f(on.)69 b(This)43 b Fy(C)1826 2957 y Fx(\003)1865 2993 y Fz(-\014eld)f(is)g (trivial)f(a)m(w)m(a)m(y)i(from)e Fu(~)h Fz(=)g(0,)i(i.e.,)g(for)d Fu(~)h Ft(6)p Fz(=)g(0,)-24 3114 y Fy(C)46 3129 y Fr(0)86 3114 y Fz(\()p Fu(R)p Fy(;)17 b Ft(B)s Fz(\))s Fu(o)426 3130 y Fs(\034)465 3112 y Fl(1)500 3130 y Fx(\012)p Fs(\013)607 3114 y Fu(R)28 b Fz(=)f Fy(S)6 b Ft(B)g Fu(o)1024 3130 y Fs(\034)1063 3112 y Fl(1)1099 3130 y Fx(\012)p Fs(\013)1206 3114 y Fu(R)1306 3086 y Ft(\030)1307 3118 y Fz(=)1411 3114 y Fy(C)1481 3129 y Fr(0)1520 3114 y Fz(\()p Fu(R)p Fy(;)17 b Ft(B)s Fz(\))s Fu(o)1860 3133 y Fs(\034)1899 3114 y Fe(~)1938 3133 y Fx(\012)p Fs(\013)2046 3114 y Fu(R)23 b Fz(and)g(k)m(er)q(\()p Fy(')2553 3129 y Fr(0)2593 3114 y Fz(\))2658 3086 y Ft(\030)2659 3118 y Fz(=)2763 3114 y Fy(C)2833 3129 y Fr(0)2873 3114 y Fz(\()p Fu(R)p Ft(nf)p Fz(0)p Ft(g)p Fy(;)17 b(C)3296 3129 y Fr(0)3334 3114 y Fz(\()p Fu(R)p Fy(;)g Ft(B)s Fz(\))s Fu(o)3674 3130 y Fs(\034)3713 3112 y Fl(1)3748 3130 y Fx(\012)p Fs(\013)3855 3114 y Fu(R)p Fz(\).)-24 3234 y(Moreo)m(v)m(er,)35 b Fy(C)502 3249 y Fr(0)541 3234 y Fz(\()p Fu(R)p Fy(;)17 b Ft(B)s Fz(\))22 b Fu(o)900 3251 y Fs(\034)939 3232 y Fl(0)974 3251 y Fx(\012)p Fs(\013)1101 3234 y Fu(R)1200 3206 y Ft(\030)1201 3238 y Fz(=)1305 3234 y Fy(S)6 b Fz(\()p Ft(B)26 b Fu(o)1577 3249 y Fs(\013)1649 3234 y Fu(R)p Fz(\))32 b(whic)m(h)i(is)f(not)g(isomorphic)g(to)g Fy(C)3025 3249 y Fr(0)3064 3234 y Fz(\()p Fu(R)p Fy(;)17 b Ft(B)s Fz(\))22 b Fu(o)3423 3251 y Fs(\034)3462 3232 y Fl(1)3497 3251 y Fx(\012)p Fs(\013)3624 3234 y Fu(R)p Fz(.)-24 3523 y Fn(3.3)136 b(Extensions)-24 3708 y Fz(Supp)s(ose)29 b(that)e(w)m(e)h(ha)m(v)m(e)g(a)f(surjectiv)m(e)j(morphism)e(b)s(et)m (w)m(een)h Fy(C)2323 3672 y Fx(\003)2363 3708 y Fz(-algebras)e Fy(q)k Fz(:)d Ft(C)6 b(!B)s Fz(.)42 b(One)28 b(then)g(sa)m(ys)g(that) -24 3828 y Ft(C)39 b Fz(is)33 b(an)g(extension)h(of)e Ft(B)k Fz(b)m(y)e(the)f(ideal)g Ft(J)45 b Fz(:=)28 b(k)m(er)q(\()p Fy(q)t Fz(\).)2005 3792 y Fr(3)-24 4056 y Fw(Example)39 b(6)e(\(Cone)h(of)f(an)h(algebra\))50 b Fz(W)-8 b(e)36 b(constructed)h(the)e(susp)s(ension)j Fy(S)6 b Ft(B)35 b Fz(=)d Fy(C)3297 4071 y Fr(0)3337 4056 y Fz(\()p Fu(R)p Fy(;)17 b Ft(B)s Fz(\))35 b(of)g(an)g(al-)-24 4177 y(gebra)f Ft(B)i Fz(in)d(Example)i(1.)45 b(Its)34 b(cone)g(is)f(giv)m(en)i(b)m(y) f Fy(C)7 b Ft(B)32 b Fz(=)c Fy(C)2191 4192 y Fr(0)2231 4177 y Fz(\()2281 4151 y(^)2269 4177 y Fu(R)o Fy(;)17 b Ft(B)s Fz(\))34 b(where)2818 4151 y(^)2806 4177 y Fu(R)29 b Fz(=)f Fu(R)23 b Ft([)g(1)p Fz(.)45 b(The)34 b(cone)f(is)h(an)-24 4297 y(extension)h(of)d Ft(B)k Fz(b)m(y)d(the)g(ideal)g Fy(S)6 b Ft(B)s Fz(,)33 b(the)g(morphism)h(b)s(eing)f Fy(q)e Fz(=)d(ev)2510 4312 y Fx(1)2584 4297 y Fz(,)33 b(the)g(ev)-5 b(aluation)33 b(at)f Ft(1)p Fz(.)-24 4525 y Fw(Example)39 b(7)e(\(Pushing)h(the)f(b)s(oundary)i(to)e(in\014nit)m (y\))49 b Fz(The)34 b(second)h(comp)s(onen)m(t)f(in)g(\()p Fy(\033)n(;)17 b(s)p Fz(\))28 b Ft(2)3769 4500 y Fz(^)3758 4525 y(\006)34 b(de-)-24 4646 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b Fy(;)217 646 y(d)268 604 y Fs(ss)342 646 y Fz(~)-54 b Fy(\034)379 661 y Fs(a)416 670 y Fl(1)456 646 y Fz(\()p Fy(f)567 630 y Fz(^)553 646 y Ft(\012)p Fz(1\)\()p Fy(a)806 661 y Fr(2)845 646 y Fz(\)\()p Fy(a)972 661 y Fr(3)1012 646 y Fz(\))83 b(=)f(\()p Fy(@)1380 661 y Fr(1)1420 646 y Fy(f)11 b Fz(\()p Fy(a)1568 661 y Fr(2)1608 646 y Fz(\)\()p Fy(a)1735 661 y Fr(3)1796 646 y Ft(\000)23 b Fy(a)1947 661 y Fr(2)1986 646 y Fy(a)2037 661 y Fr(1)2077 646 y Fz(\))f Ft(\000)h Fy(a)2288 661 y Fr(1)2327 646 y Fy(@)2378 661 y Fr(2)2418 646 y Fy(f)11 b Fz(\()p Fy(a)2566 661 y Fr(2)2606 646 y Fz(\)\()p Fy(a)2733 661 y Fr(3)2794 646 y Ft(\000)23 b Fy(a)2945 661 y Fr(2)2985 646 y Fy(a)3036 661 y Fr(1)3075 646 y Fz(\)\))3182 630 y(^)3168 646 y Ft(\012)17 b Fy(e)3307 661 y Fr(1)1291 791 y Fz(+)22 b Fy(@)1440 806 y Fr(2)1480 791 y Fy(f)11 b Fz(\()p Fy(a)1628 806 y Fr(2)1668 791 y Fz(\)\()p Fy(a)1795 806 y Fr(3)1856 791 y Ft(\000)23 b Fy(a)2007 806 y Fr(2)2047 791 y Fy(a)2098 806 y Fr(1)2137 791 y Fz(\))2189 776 y(^)2175 791 y Ft(\012)q Fy(e)2298 806 y Fr(2)2365 791 y Fy(:)-24 1000 y Fz(F)-8 b(urthermore,)48 b(~)-54 b Fy(\034)51 b Fz(lea)m(v)m(es)42 b(the)e(graded)g(trace)h(T)-8 b(r)1812 958 y Fs(ss)1921 1000 y Fz(in)m(v)j(arian)m(t,)43 b(b)s(ecause)e(of)j(~)-54 b Fy(\034)2895 1015 y Fs(a)2932 1024 y Fl(1)2972 1000 y Fz(\(1)3073 985 y(^)3059 1000 y Ft(\012)p Fy(e)3181 1015 y Fr(1)3221 1000 y Fy(e)3266 1015 y Fr(2)3305 1000 y Fz(\))40 b(=)h(1)3563 985 y(^)3549 1000 y Ft(\012)p Fy(e)3671 1015 y Fr(1)3710 1000 y Fy(e)3755 1015 y Fr(2)3835 1000 y Fz(and)-24 1121 y(the)h(translation)e(in)m(v)-5 b(ariance)42 b(of)e(the)i(Leb)s(esgue)g(measure)g(on)e Fu(R)2452 1084 y Fr(2)2492 1121 y Fz(.)68 b(In)41 b(order)f(to)h(apply) g(Prop)s(osition)g(5,)-24 1241 y(w)m(e)36 b(sho)m(w)g(that)e(the)h (2-cycle)h(\()p Fu(C)1184 1205 y Fs(ss)1253 1241 y Fy(;)17 b(d)1348 1205 y Fs(ss)1417 1241 y Fy(;)g Fz(T)-8 b(r)1561 1199 y Fs(ss)1630 1241 y Fz(\))35 b(satis\014es)h(the)f(uniform)g(b)s (ound)g(\(18\))e(w.r.t.)j(the)f(dense)h(sub-)-24 1361 y(algebra)28 b Fy(C)390 1325 y Fr(1)383 1386 y Fs(c)429 1361 y Fz(\()p Fu(R)539 1325 y Fr(2)579 1361 y Fy(;)17 b Fu(C)p Fz(\))27 b Ft(\032)h Fy(S)6 b(S)g Fu(C)27 b Fz(giv)m(en)i(b)m(y)f(con)m(tin)m(uously)i(di\013eren)m(tiable)g (functions)e(with)g(compact)h(supp)s(ort.)-24 1482 y(One)36 b(\014nds)g(for)e(the)h(norms)h Fy(p)p Fz(\()p Fy(f)1173 1497 y Fr(1)1212 1482 y Fy(;)17 b(f)1304 1497 y Fr(2)1344 1482 y Fz(\))34 b(\()p Fo(cf.)51 b Fz(De\014nition)35 b(3\),)h(using)f Fy(@)2519 1497 y Fs(a)2556 1506 y Fl(2)2601 1482 y Fz(~)-54 b Fy(\034)2638 1497 y Fs(t)2668 1482 y Fz(\()p Fy(f)2754 1497 y Fr(1)2793 1482 y Fz(\))32 b(=)37 b(~)-54 b Fy(\034)3013 1497 y Fs(t)3043 1482 y Fz(\()p Fy(@)3132 1497 y Fr(1)3171 1482 y Fy(f)3219 1497 y Fr(1)3259 1482 y Fz(\))24 b Ft(\000)g Fy(t)e Fz(~)-54 b Fy(\034)3516 1497 y Fs(t)3546 1482 y Fz(\()p Fy(@)3635 1497 y Fr(2)3675 1482 y Fy(f)3723 1497 y Fr(1)3762 1482 y Fz(\))35 b(and)-24 1602 y Fy(@)27 1617 y Fs(a)64 1626 y Fl(3)109 1602 y Fz(~)-54 b Fy(\034)146 1617 y Fs(t)176 1602 y Fz(\()p Fy(f)262 1617 y Fr(1)302 1602 y Fz(\))27 b(=)33 b(~)-54 b Fy(\034)513 1617 y Fs(t)543 1602 y Fz(\()p Fy(@)632 1617 y Fr(2)672 1602 y Fy(f)720 1617 y Fr(1)759 1602 y Fz(\),)552 1811 y Fy(p)p Fz(\()5 b(~)-54 b Fy(\034)681 1826 y Fs(t)711 1811 y Fz(\()p Fy(f)797 1826 y Fr(1)837 1811 y Fz(\))p Fy(;)17 b(f)967 1826 y Fr(2)1006 1811 y Fz(\))83 b Ft(\024)g Fz(\()p Ft(k)5 b Fz(~)-54 b Fy(\034)1417 1826 y Fs(t)1447 1811 y Fz(\()p Fy(@)1536 1826 y Fr(1)1576 1811 y Fy(f)1624 1826 y Fr(1)1663 1811 y Fz(\))p Ft(k)17 b(k)p Fy(@)1869 1826 y Fr(2)1908 1811 y Fy(f)1956 1826 y Fr(2)1996 1811 y Ft(k)50 b Fz(+)f Ft(k)5 b Fz(~)-54 b Fy(\034)2313 1826 y Fs(t)2343 1811 y Fz(\()p Fy(@)2432 1826 y Fr(2)2472 1811 y Fy(f)2520 1826 y Fr(1)2559 1811 y Fz(\))p Ft(k)17 b(k)p Fy(@)2765 1826 y Fr(1)2805 1811 y Fy(f)2853 1826 y Fr(2)2892 1811 y Ft(k)p Fz(\))f Ft(j)p Fz(supp)q(\()p Fy(f)3311 1826 y Fr(2)3351 1811 y Fz(\))p Ft(j)1287 1957 y Fz(+)28 b Ft(k)5 b Fz(~)-54 b Fy(\034)1483 1972 y Fs(t)1513 1957 y Fz(\()p Fy(@)1602 1972 y Fr(2)1641 1957 y Fy(f)1689 1972 y Fr(1)1729 1957 y Fz(\))p Ft(k)17 b(k)p Fy(@)1935 1972 y Fr(2)1974 1957 y Fy(f)2022 1972 y Fr(2)2062 1957 y Ft(k)27 b(j)p Fy(t)p Ft(j)g(j)p Fz(supp)q(\()5 b(~)-54 b Fy(\034)2566 1972 y Fs(t)2596 1957 y Fy(f)2644 1972 y Fr(1)2684 1957 y Fz(\))22 b Ft(\\)g Fz(supp)q(\()p Fy(f)3119 1972 y Fr(2)3159 1957 y Fz(\))p Ft(j)27 b Fy(:)-24 2166 y Fz(Here)45 b Ft(k)p Fy(:)p Ft(k)e Fz(is)i(the)f(suprem)m(um)i (norm)e(on)g Fy(S)6 b(S)g Fu(C)p Fz(.)76 b(Since)45 b Ft(j)p Fy(t)p Ft(j)i(j)p Fz(supp)q(\()5 b(~)-54 b Fy(\034)2596 2181 y Fs(t)2626 2166 y Fy(f)2674 2181 y Fr(1)2713 2166 y Fz(\))30 b Ft(\\)g Fz(supp)q(\()p Fy(f)3164 2181 y Fr(2)3204 2166 y Fz(\))p Ft(j)43 b Fz(is)h(b)s(ounded)h(in)f Fy(t)-24 2286 y Fz(for)d(an)m(y)g(t)m(w)m(o)g(compactly)h(supp)s(orted) g(functions)f Fy(f)1938 2301 y Fr(1)1978 2286 y Fy(;)17 b(f)2070 2301 y Fr(2)2109 2286 y Fz(,)43 b(w)m(e)f(obtain)e(the)h (desired)i(result,)g(namely)f(that)-24 2407 y Fy(p)p Fz(\()5 b(~)-54 b Fy(\034)105 2422 y Fs(t)135 2407 y Fz(\()p Fy(f)221 2422 y Fr(1)261 2407 y Fz(\))p Fy(;)17 b(f)391 2422 y Fr(2)430 2407 y Fz(\))32 b(is)h(b)s(ounded)h(in)f Fy(t)p Fz(.)123 2527 y(Hence)e(w)m(e)g(are)f(in)g(a)f(p)s(osition)i(to) e(apply)i(Prop)s(osition)f(5)f(from)h(whic)m(h)h(w)m(e)g(then)g(obtain) f(a)f(3-cycle)i(o)m(v)m(er)-24 2647 y Fy(L)42 2611 y Fr(1)82 2647 y Fz(\()p Fu(R)p Fy(;)17 b(S)6 b(S)g Fu(C)p Fy(;)21 b Fz(~)-53 b Fy(\034)11 b Fz(\))32 b(giv)m(en)i(b)m(y)805 2774 y Fp(\022)878 2914 y Fy(L)944 2873 y Fr(1)984 2914 y Fz(\()p Fu(R)p Fy(;)17 b(S)6 b(S)g Fu(C)1356 2899 y Fz(^)1342 2914 y Ft(\012)o Fz(\003)p Fu(C)1558 2873 y Fr(2)1597 2914 y Fy(;)22 b Fz(~)-54 b Fy(\034)12 b Fz(\))1747 2899 y(^)1733 2914 y Ft(\012)p Fz(\003)p Fu(C)p Fy(;)17 b(d)2045 2873 y Fs(ss)2049 2939 y Fr(~)-39 b Fs(\034)2114 2914 y Fy(;)2158 2778 y Fp(Z)2213 3004 y Fk(R)2261 2985 y Fl(2)2316 2914 y Fy(da)2418 2929 y Fr(2)2457 2914 y Fy(da)2559 2929 y Fr(3)2615 2914 y Fz(T)-8 b(r)17 b(ev)2827 2929 y Fr(0)2888 2914 y Ft(\012)23 b Fy({)3019 2774 y Fp(\023)3137 2914 y Fy(:)-24 3175 y Fz(Its)39 b(c)m(haracter)g(is)g (the)g(canonical)f(co)s(cycle)i(of)e(the)h(Heisen)m(b)s(erg)h(group)e (algebra.)60 b(Prop)s(osition)38 b(5)g(insures)-24 3295 y(that)43 b(this)h(c)m(haracter)g(is)g(a)f(3-trace)g(on)g(the)g(Banac)m (h)h(sub-algebra)g Fy(L)2653 3259 y Fr(1)2692 3295 y Fz(\()p Fu(R)p Fy(;)17 b(S)6 b(S)g Fu(C)p Fy(;)21 b Fz(~)-53 b Fy(\034)11 b Fz(\))43 b(of)g(the)g(Heisen)m(b)s(erg)-24 3416 y(group)33 b(algebra,)f(but)h(this)g(is)g(su\016cien)m(t)i(for)d (our)h(purp)s(oses.)-24 3632 y Fw(Example)39 b(14)49 b Fz(If)25 b(\(\012)p Fy(;)17 b(d;)937 3552 y Fp(R)1003 3632 y Fz(\))26 b(is)g(an)g Fy(\013)q Fz(-in)m(v)-5 b(arian)m(t)25 b Fy(n)p Fz(-cycle)i(o)m(v)m(er)g Ft(B)s Fz(,)g(then)g(the)f(ab)s(o)m (v)m(e)g(construction)h(straigh)m(t-)-24 3752 y(forw)m(ardly)h (generalizes)h(to)e(the)g Fy(C)1243 3716 y Fx(\003)1283 3752 y Fz(-\014eld)g Fy(S)6 b(S)g Ft(B)14 b Fu(o)1813 3767 y Fr(~)-39 b Fs(\034)9 b Fx(\012)p Fs(\013)1964 3752 y Fu(R)27 b Fz(from)g(Example)h(5)f(and)g(w)m(e)i(obtain)e(the)g Fy(n)11 b Fz(+)g(3-cycle)676 3875 y Fp(\022)750 4016 y Fy(L)816 3975 y Fr(1)856 4016 y Fz(\()p Fu(R)p Fy(;)17 b(S)6 b(S)g Fz(\012)1226 4001 y(^)1212 4016 y Ft(\012)p Fz(\003)p Fu(C)1429 3975 y Fr(2)1468 4016 y Fy(;)22 b Fz(~)-54 b Fy(\034)33 b Ft(\012)23 b Fy(\013)p Fz(\))1801 4001 y(^)1787 4016 y Ft(\012)q Fz(\003)p Fu(C)p Fy(;)17 b(d)2100 3975 y Fs(ss)2104 4040 y Fr(~)-39 b Fs(\034)7 b Fx(\012)p Fs(\013)2242 4016 y Fy(;)2286 3880 y Fp(Z)2341 4106 y Fk(R)2389 4087 y Fl(2)2445 4016 y Fy(da)2547 4031 y Fr(2)2586 4016 y Fy(da)2688 4031 y Fr(3)2744 3880 y Fp(Z)2860 4016 y Fz(ev)2955 4031 y Fr(0)3017 4016 y Ft(\012)22 b Fy({)3147 3875 y Fp(\023)3265 4016 y Fy(:)-24 4287 y Fz(whose)32 b(c)m(haracter)f(is)g(an)f Fy(n)18 b Fz(+)f(3-trace)30 b(on)g(the)h(Banac)m(h)f(sub-algebra)h Fy(L)2635 4251 y Fr(1)2675 4287 y Fz(\()p Fu(R)p Fy(;)17 b(S)6 b(S)g Ft(B)s Fy(;)21 b Fz(~)-53 b Fy(\034)28 b Ft(\012)18 b Fy(\013)q Fz(\).)42 b(The)31 b(details)h(of)-24 4407 y(the)i(pro)s(of)d(that)i(the)g(in)m(termediate)i Fy(n)22 b Fz(+)g(2-cycle)34 b(is)k(~)-54 b Fy(\034)34 b Ft(\012)22 b Fy(\013)q Fz(-in)m(v)-5 b(arian)m(t)33 b(are)g(w)m(ork)m(ed)h(out)f (as)g(ab)s(o)m(v)m(e)g(and)g(w)m(e)-24 4527 y(only)g(indicate)h(ho)m(w) f(to)f(sho)m(w)i(the)f(b)s(ound)g(\(18\))f(for)g(the)h Fy(n)22 b Fz(+)g(2-cycle)1192 4654 y Fp(\022)1266 4794 y Fy(S)6 b(S)g Fz(\012)1482 4779 y(^)1468 4794 y Ft(\012)p Fz(\003)p Fu(C)1685 4753 y Fr(2)1724 4794 y Fy(;)17 b(d)1819 4753 y Fs(ss)1888 4794 y Fy(;)1932 4658 y Fp(Z)1987 4884 y Fk(R)2035 4865 y Fl(2)2091 4794 y Fy(da)2193 4809 y Fr(2)2232 4794 y Fy(da)2334 4809 y Fr(3)2390 4658 y Fp(Z)2506 4794 y Ft(\012)g Fy({)2631 4654 y Fp(\023)2749 4794 y Fy(:)-24 5065 y Fz(F)-8 b(or)32 b(that)g(consider)i(the)f(dense)h (sub-algebra)f Ft(V)1766 5029 y Fl(\014n)1867 5065 y Fz(=)1971 4990 y 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2694 y Fz(to)f(b)s(e)-24 2814 y(closed)34 b(and)f(therefore)g(extends)i(to)d(the)h(case)g(of)f(c)m(hains.)123 2982 y(An)39 b(example)h(of)e(an)h(un)m(b)s(ounded)h(c)m(hain)f(for)g (a)f(comm)m(utativ)m(e)j(algebra)d Fy(C)7 b Fz(\()p Fy(M)j Fz(\))40 b(of)e(functions)i(o)m(v)m(er)f(a)-24 3102 y(compact)h (manifold)e(is)h(obtained)g(when)h(one)f(lo)s(oks)g(at)f(the)h(algebra) f(of)g(exterior)h(forms)g(with)g(its)g(usual)-24 3223 y(di\013eren)m(tial)f(and)e(in)m(tegration)g(structure,)i(but)e Fy(M)47 b Fz(has)36 b(a)g(b)s(oundary)g Fy(@)5 b(M)10 b Fz(.)56 b(The)37 b(map)f Fy(r)i Fz(is)f(then)f(simply)-24 3343 y(the)42 b(restriction)g(to)f(the)h(b)s(oundary)f(and)g Fy(@)5 b Fz(\012)44 b(=)e(\012\()p Fy(@)5 b(M)10 b Fz(\).)71 b(In)42 b(this)g(con)m(text)g(Stok)m(es')h(Theorem)f(relates)-24 3463 y(in)m(tegration)k(of)e(exact)i(dim\()p Fy(M)10 b Fz(\)-forms)45 b(o)m(v)m(er)h Fy(M)56 b Fz(to)44 b(the)i(in)m (tegration)f(of)f(a)h(form)f(o)m(v)m(er)i(the)g(b)s(oundary)-24 3584 y Fy(@)5 b(M)10 b 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Ft(B)s Fz(\))g(whic)m(h)j(is)e(closed)g(under)h(holomorphic)f (functional)g(calculus.)123 5362 y(The)g(follo)m(wing)g(example)h(pla)m (ys)g(the)f(analog)f(in)h(cyclic)h(cohomology)f(of)f(the)h(ENN-map)g (in)g(\(12\).)1935 5611 y(24)p eop end %%Page: 25 25 TeXDict begin 25 24 bop -24 171 a Fw(Example)39 b(15)49 b Fz(Consider)32 b(the)f(con)m(tin)m(uous)h Fy(C)1734 134 y Fx(\003)1774 171 y Fz(-\014eld)e Fy(S)6 b(S)g Ft(B)22 b Fu(o)2315 186 y Fr(~)-39 b Fs(\034)8 b Fx(\012)p Fs(\013)2473 171 y Fu(R)30 b Fz(from)h(Example)h(5)e(together)h(with)g(an)-24 291 y Fy(\013)37 b Fz(in)m(v)-5 b(arian)m(t)37 b Fy(n)p Fz(-cycle)g(\(\012)p Fy(;)17 b(d;)1070 211 y Fp(R)1136 291 y Fz(\))36 b(o)m(v)m(er)h Ft(B)s Fz(.)54 b(In)36 b(Example)i(14)d(w)m(e)i(ha)m(v)m(e)g(constructed)h(an)e Fy(n)25 b Fz(+)f(3-cycle)37 b(for)e(a)-24 411 y(dense)k(Banac)m(h)f (sub-algebra)g(of)f(the)g(\014eld.)59 b(No)m(w)38 b(restrict)g(this)g Fy(C)2518 375 y Fx(\003)2557 411 y Fz(-\014eld)g(to)f(the)h(in)m(terv) -5 b(al)38 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Fk(~)1807 5074 y Fl(0)1847 5050 y Fy(;)17 b(x)1946 5065 y Fk(~)1984 5074 y Fl(1)2023 5050 y Fz(\))33 b(is)h(a)g(pro)5 b(jection)34 b(\(unitary\))g(in)g(that)g(algebra.)47 b(By)-24 5170 y(Prop)s(osition)27 b(7)f(\()p Fy(x)663 5185 y Fk(~)701 5194 y Fl(0)740 5170 y Fy(;)17 b(x)839 5185 y Fk(~)877 5194 y Fl(1)916 5170 y Fz(\))26 b(therefore)h(pairs)g (trivially)g(with)g(the)g(c)m(haracter)g(of)e(the)i(b)s(oundary)g(of)e (the)i(c)m(hain,)-24 5291 y(i.e.)44 b(0)28 b(=)f Ft(h)p Fz(#)444 5313 y Fs(\034)483 5291 y Fe(~)517 5306 y Fl(1)556 5313 y Fx(\012)p Fs(\013)661 5291 y Fy(\021)713 5254 y Fs(s)749 5291 y Fy(;)17 b(x)848 5306 y Fk(~)886 5315 y Fl(1)926 5291 y Ft(i)k(\000)i(h)p Fz(#)1206 5313 y Fs(\034)1245 5291 y Fe(~)1279 5306 y Fl(0)1318 5313 y Fx(\012)p Fs(\013)1423 5291 y Fy(\021)1475 5254 y Fs(s)1511 5291 y Fy(;)17 b(x)1610 5306 y Fk(~)1648 5315 y Fl(0)1687 5291 y Ft(i)28 b Fz(.)2137 b Fd(2)1935 5611 y Fz(27)p eop end %%Page: 28 28 TeXDict begin 28 27 bop -24 171 a Fw(Example)39 b(16)49 b Fz(Lemma)35 b(1\(i\))g(is)g(an)g(application)g(of)f(this)i(prop)s (osition)f(to)f Ft(B)h Fz(=)c Fu(C)p Fz(,)36 b Fy(\013)c Fz(=)f(id)k(and)g Fy(\021)g Fz(=)d(T)-8 b(r.)-24 291 y(Then)29 b Fy(S)6 b Ft(B)16 b Fu(o)450 306 y Fs(\034)489 287 y Fj(\015)530 306 y Fx(\012)p Fl(id)652 291 y Fu(R)752 263 y Ft(\030)753 295 y Fz(=)857 291 y Fy(C)934 255 y Fx(\003)927 316 y Fs(\015)974 291 y Fz(\()p Fy(D)1093 306 y Fr(1)1132 291 y Fy(;)h(D)1257 306 y Fr(2)1296 291 y Fz(\))28 b(\(see)h(Example)g(2\))f(and)g(c)m(h)g(=)g Ft(\000)p Fy({)p Fz(#)2668 306 y Fs(\034)2707 287 y Fj(\015)2748 306 y Fx(\012)p Fl(id)2858 291 y Fz(T)-8 b(r)2958 249 y Fs(s)2995 291 y Fz(.)42 b(Example)29 b(4)f(establishes)-24 411 y(that)38 b(the)g Fy(C)443 375 y Fx(\003)436 436 y Fs(\015)482 411 y Fz(\()p Fy(D)601 426 y Fr(1)641 411 y Fy(;)17 b(D)766 426 y Fr(2)805 411 y Fz(\))37 b(for)h(di\013eren)m(t) g Fy(\015)43 b Fz(constitute)c(a)e Fy(C)2142 375 y Fx(\003)2182 411 y Fz(-\014eld)h(and,)h(applying)f(the)g(ab)s(o)m(v)m(e)h(prop)s (osition)-24 532 y(to)33 b(the)g(section)g Fy(P)652 547 y Fs(\015)t(D)750 555 y Fj(\015)827 532 y Fz(yields)h Ft(h)p Fz(c)m(h)q Fy(;)17 b Fz(\005)1352 547 y Fs(\015)1396 532 y Ft(i)27 b Fz(=)h Ft(h)p Fz(c)m(h)1699 547 y Fs(g)1740 532 y Fy(;)17 b(b)p Ft(i)32 b Fz(as)h([\005)2116 547 y Fs(\015)2160 532 y Fz(])2187 547 y Fr(0)2255 532 y Fz(=)27 b Fy(\026)2417 547 y Fr(0)2456 532 y Fz([)p Fy(b)p Fz(])2551 547 y Fr(0)2624 532 y Fz(\()p Fo(cf.)44 b Fz(Example)34 b(10\).)-24 736 y Fw(Theorem)k(4)49 b Fv(L)-5 b(et)37 b Fy(x)g Fv(b)-5 b(e)36 b(a)g(r)-5 b(epr)g(esentative)36 b(for)g(an)g(element)g(in)g Fy(K)2512 751 y Fs(i)2540 736 y Fz(\()p Ft(B)27 b Fu(o)2747 751 y Fs(\013)2820 736 y Fu(R)p Fz(\))36 b Fv(and)g Fy(\021)k Fv(b)-5 b(e)36 b(an)g Fy(\013)q Fv(-invariant)-24 856 y(cyclic)f(c)-5 b(o)g(cycle)34 b(over)h Ft(B)s Fv(.)45 b(Then)1300 1055 y Ft(h)p Fz(#)1420 1070 y Fs(\013)1470 1055 y Fy(\021)t(;)17 b(x)p Ft(i)55 b Fz(=)g Ft(\000h)p Fz(#)2043 1070 y Fs(\034)8 b Fx(\012)p Fs(\013)2187 1055 y Fy(\021)2239 1013 y Fs(s)2275 1055 y Fy(;)17 b(@)2370 1070 y Fs(i)2399 1055 y Fz(\()p Fy(x)p Fz(\))p Ft(i)35 b Fv(,)1185 b Fz(\(24\))-24 1253 y Fv(wher)-5 b(e)31 b Fy(@)299 1268 y Fr(0)367 1253 y Fz(=)c(exp)i(:)f Fy(K)785 1268 y Fr(0)824 1253 y Fz(\()p Ft(B)17 b Fu(o)1021 1268 y Fs(\013)1085 1253 y Fu(R)p Fz(\))28 b Ft(!)f Fy(K)1433 1268 y Fr(1)1473 1253 y Fz(\()p Fy(S)6 b Ft(B)17 b Fu(o)1736 1268 y Fs(\034)8 b Fx(\012)p Fs(\013)1894 1253 y Fu(R)p Fz(\))31 b Fv(and)f Fy(@)2271 1268 y Fr(1)2339 1253 y Fz(=)d(ind)i(:)e Fy(K)2743 1268 y Fr(1)2783 1253 y Fz(\()p Ft(B)17 b Fu(o)2980 1268 y Fs(\013)3044 1253 y Fu(R)p Fz(\))28 b Ft(!)f Fy(K)3392 1268 y Fr(0)3431 1253 y Fz(\()p Fy(S)6 b Ft(B)18 b Fu(o)3695 1268 y Fs(\034)8 b Fx(\012)p Fs(\013)3852 1253 y Fu(R)p Fz(\))p Fv(.)-24 1457 y Fw(Pro)s(of:)30 b Fz(Applying)h(Prop)s(osition) f(8)g(to)g(the)g(p)s(oin)m(ts)h Fu(~)c Fz(=)h(0)h(and)h Fu(~)e Fz(=)f(1)j(and)g(the)h(Prop)s(osition)f(1,)g(it)g(su\016ces)-24 1577 y(to)j(sho)m(w)g(that)g Ft(h)p Fz(#)669 1592 y Fs(\013)718 1577 y Fy(\021)t(;)17 b(x)p Ft(i)27 b Fz(=)h Ft(\000h)p Fz(#)1236 1592 y Fl(id)p Fx(\012)p Fs(\013)1409 1577 y Fy(\021)1461 1541 y Fs(s)1497 1577 y Fy(;)17 b(@)1592 1592 y Fs(i)1621 1577 y Fz(\()p Fy(x)p Fz(\))p Ft(i)32 b Fz(where)i(no)m(w)f Fy(@)2359 1592 y Fr(0)2427 1577 y Fz(=)27 b Fy(\014)38 b Fz(and)33 b Fy(@)2864 1592 y Fr(1)2932 1577 y Fz(=)27 b(\002.)123 1698 y(Let)36 b(us)i(lo)s(ok)e (here)h(at)f(the)h(case)h Fy(i)c Fz(=)h(0)h(in)h(whic)m(h)h Fy(x)e Fz(is)h(represen)m(ted)j(b)m(y)d(a)f(pro)5 b(jection)37 b Fy(p)p Fz(.)56 b(Then)37 b Fy(\014)6 b Fz([)p Fy(p)p Fz(])-24 1818 y(is)34 b(represen)m(ted)h(b)m(y)e(the)g(unitary)g Fy(Gp)22 b Fz(+)g(1)33 b(where)h Fy(G)p Fz(\()p Fy(s)p Fz(\))27 b(=)h Fy(e)2225 1782 y Fr(2)p Fs(\031)r({\037)p Fr(\()p Fs(s)p Fr(\))2484 1818 y Ft(\000)23 b Fz(1.)43 b(Since)34 b Fy(G)e Fz(comm)m(utes)j(with)e Fy(p)p Fz(,)g(the)-24 1939 y(calculation)41 b(is)f(simple.)67 b(If)40 b Fy(\021)k Fz(is)c(the)h(c)m(haracter)f(of)g(the)g(2)p Fy(k)30 b Fz(+)d(1)40 b(cycle)h(\(\012)p Fy(;)17 b(d;)2967 1858 y Fp(R)3033 1939 y Fz(\),)42 b(then)e(#)3450 1954 y Fl(id)p Fx(\012)p Fs(\013)3623 1939 y Fy(\021)3675 1902 y Fs(s)3751 1939 y Fz(is)h(the)-24 2059 y(c)m(haracter)31 b(of)e(\(\012)615 2023 y Fs(s)615 2084 y(\013)665 2059 y Fy(;)17 b(d)760 2023 y Fs(s)760 2084 y(\013)809 2059 y Fy(;)853 1979 y Fp(R)900 2094 y Fk(R)968 1979 y Fp(R)1052 2059 y Fz(ev)1146 2074 y Fr(0)1186 2059 y Fz(\))29 b(and)h(w)m(e)h(write)g(again)e Fy(d)2136 2023 y Fs(s)2136 2084 y(\013)2213 2059 y Fz(=)e Fy(\016)2359 2074 y Fr(1)2415 2059 y Fz(+)16 b Fy(d)2558 2074 y Fs(\013)2638 2059 y Fz(where)31 b Fy(\016)2960 2074 y Fr(1)2999 2059 y Fz(\()p Fy(f)3110 2044 y Fz(^)3096 2059 y Ft(\012)p Fz(1\))d(=)f Fy(@)3442 2074 y Fs(s)3480 2059 y Fy(f)3553 2044 y Fz(^)3539 2059 y Ft(\012)p Fy(e)3661 2074 y Fr(2)3700 2059 y Fz(.)43 b(Then)-24 2179 y Fy(d)27 2143 y Fs(s)27 2204 y(\013)77 2179 y Fz(\()p Fy(Gp)p Fz(\))27 b(=)h(\()p Fy(@)499 2194 y Fs(s)536 2179 y Fy(G)p Fz(\))p Fy(p)714 2164 y Fz(^)700 2179 y Ft(\012)p Fy(e)822 2194 y Fr(2)883 2179 y Fz(+)22 b Fy(Gd)1109 2194 y Fs(\013)1158 2179 y Fy(p)32 b Fz(so)g(that,)h(using)g Fy(p)p Fz(\()p Fy(d)1990 2194 y Fs(\013)2039 2179 y Fy(p)p Fz(\))2126 2143 y Fs(m)2192 2179 y Fy(p)28 b Fz(=)f Fy(p)p Fz(\()p Fy(d)2510 2194 y Fs(\013)2559 2179 y Fy(p)p Fz(\))2646 2143 y Fs(m)2745 2179 y Fz(for)32 b(ev)m(en)i Fy(m)e Fz(and)h Fy(p)p Fz(\()p Fy(d)3561 2194 y Fs(\013)3610 2179 y Fy(p)p Fz(\))3697 2143 y Fs(m)3763 2179 y Fy(p)28 b Fz(=)f(0)-24 2300 y(for)32 b(o)s(dd)h Fy(m)p Fz(,)g(one)g(gets)438 2541 y Ft(h)p Fz(#)558 2556 y Fl(id)p Fx(\012)p Fs(\013)714 2541 y Fy(\021)766 2500 y Fs(s)803 2541 y Fy(;)17 b(Gp)k Fz(+)h(1)p Ft(i)83 b Fz(=)g Fy(c)1464 2556 y Fr(2)p Fs(k)r Fr(+3)1648 2406 y Fp(Z)1704 2631 y Fk(R)1772 2406 y Fp(Z)1889 2541 y Fz(ev)1984 2556 y Fr(0)2040 2431 y Fp(\020)2121 2516 y Fz(\026)2099 2541 y Fy(Gp)17 b(d)2293 2500 y Fs(s)2293 2566 y(\013)2342 2541 y Fz(\()p Fy(Gp)p Fz(\))2561 2460 y Fp(\000)2606 2541 y Fy(d)2657 2500 y Fs(s)2657 2566 y(\013)2706 2541 y Fz(\()2766 2516 y(\026)2744 2541 y Fy(Gp)p Fz(\))p Fy(d)2959 2500 y Fs(s)2959 2566 y(\013)3008 2541 y Fz(\()p Fy(Gp)p Fz(\))3210 2460 y Fp(\001)3255 2483 y Fs(k)r Fr(+1)3388 2431 y Fp(\021)1263 2800 y Fz(=)83 b Ft(\000)28 b Fz(\()p Fy(k)d Fz(+)d(2\))p Fy(c)1868 2815 y Fr(2)p Fs(k)r Fr(+3)2052 2664 y Fp(Z)2108 2890 y Fk(R)2177 2800 y Fy(G)2254 2759 y Fx(0)2277 2800 y Fy(G)2354 2759 y Fs(k)r Fr(+1)2509 2775 y Fz(\026)2487 2800 y Fy(G)2564 2759 y Fs(k)r Fr(+2)2713 2664 y Fp(Z)2829 2800 y Fz(ev)2924 2815 y Fr(0)2980 2719 y Fp(\000)3026 2800 y Fy(p)p Fz(\()p Fy(d)3164 2815 y Fs(\013)3213 2800 y Fy(p)p Fz(\))3300 2759 y Fr(2)p Fs(k)r Fr(+2)3468 2719 y Fp(\001)1263 2998 y Fz(=)83 b Ft(\000)28 b(h)p Fz(#)1647 3013 y Fs(\013)1696 2998 y Fy(\021)t(;)17 b(p)p Ft(i)p Fy(:)-24 3207 y Fz(since)215 3126 y Fp(R)262 3241 y Fk(R)331 3207 y Fy(G)408 3170 y Fx(0)431 3207 y Fy(G)508 3170 y Fs(k)r Fr(+1)663 3181 y Fz(\026)641 3207 y Fy(G)718 3170 y Fs(k)r Fr(+2)879 3207 y Fz(=)27 b(2)p Fy(\031)t({)1243 3159 y Fr(\(2)p Fs(k)r Fr(+3\)!)p 1159 3184 407 4 v 1159 3241 a(\()p Fs(k)r Fr(+1\)!\()p Fs(k)r Fr(+2\)!)1592 3207 y Fz(.)123 3374 y(The)43 b(case)h Fy(i)h Fz(=)g(1)d(is)h(more)g (in)m(v)m(olv)m(ed)i(and)e(since)h(w)m(e)f(do)g(not)f(need)i(it)f(for)f (our)g(application)h(to)g(the)-24 3495 y(quan)m(tum)34 b(Hall)f(e\013ect,)g(w)m(e)h(refer)f(the)g(reader)g(to)f([P83])h(for)f (this)h(result.)1228 b Fd(2)-24 3780 y Fn(6.3)136 b(P)l(erio)t(dicit)l (y)46 b(in)f(cyclic)g(cohomology)-24 3965 y Fz(Under)d(pairing)f(with)g Fy(K)7 b Fz(-theory)-8 b(,)44 b(cyclic)e(cohomology)f(b)s(eha)m(v)m(es) i(lik)m(e)f(a)f(p)s(erio)s(dic)g(cohomology)g(theory)-8 b(.)-24 4085 y(This)40 b(means,)g(in)f(particular,)h(that)e(there)h (exists)h(a)e(map)g(\(denoted)h Fy(S)44 b Fz(in)39 b([C94]\))f(that)g (assigns)h(to)f(eac)m(h)-24 4206 y(cyclic)h Fy(n)p Fz(-co)s(cycle)f(a)f (cyclic)i Fy(n)25 b Fz(+)h(2-co)s(cycle)37 b(whic)m(h)i(pairs)e(in)h (the)f(same)h(w)m(a)m(y)h(with)e Fy(K)7 b Fz(-theory)-8 b(.)57 b(This)39 b(is)e(a)-24 4326 y(w)m(ell-kno)m(wn)e(result,)f(but)f (for)f(the)h(con)m(v)m(enience)i(of)d(the)h(reader)h(w)m(e)f(presen)m (t)h(a)f(detailed)g(pro)s(of.)123 4494 y(Recall)38 b(from)g (De\014nition)g(1)g(that)g(w)m(e)h(extend)g(a)f(co)s(cycle)i Fy(\021)h Fz(from)d Ft(B)j Fz(to)d(matrix)h(algebras)f(o)m(v)m(er)h Ft(B)i Fz(or)-24 4614 y(to)c Ft(K)27 b(\012)e(B)41 b Fz(b)m(y)d(the)f(op)s(erator)g(trace)g(on)g(the)g(left)h(factor,)f(T)-8 b(r)26 b Ft(\012)g Fy(\021)t Fz(.)56 b(As)38 b(w)m(e)g(shall)g(see)g(b) s(elo)m(w,)h(there)f(is)f(an)-24 4734 y(isomorphism)d Fy(S)6 b Ft(B)23 b Fu(o)777 4749 y Fs(\034)8 b Fx(\012)p Fs(\013)942 4734 y Fu(R)1041 4707 y Ft(\030)1042 4739 y Fz(=)1146 4734 y Ft(K)22 b(\012)f(B)35 b Fz(for)c(an)m(y)i Fu(R)p Fz(-action)e Fy(\013)h Fz(on)g Ft(B)j Fz(and)d(under)g(this)h (isomorphism)g(T)-8 b(r)21 b Ft(\012)f Fy(\021)-24 4855 y Fz(is)33 b(iden)m(ti\014ed)i(\(Lemma)e(2\))f(with)h(the)g(c)m (haracter)h Fy(\021)1873 4819 y Fs(e)1942 4855 y Fz(of)e(the)h Fy(n)p Fz(-cycle)h(\()p Fy(S)6 b Fz(\012)22 b Fu(o)2826 4870 y Fs(\034)8 b Fx(\012)p Fs(\013)2992 4855 y Fu(R)p Fy(;)17 b(d)3159 4819 y Fs(e)3196 4855 y Fy(;)3240 4775 y Fp(R)3306 4801 y Fs(e)3343 4855 y Fz(\),)700 5110 y(\()p Fy(d)789 5069 y Fs(e)826 5110 y Fy(f)11 b Fz(\)\()p Fy(x)p Fz(\)\()p Fy(s)p Fz(\))55 b(=)g Fy(d)p Fz(\()p Fy(f)11 b Fz(\()p Fy(x)p Fz(\)\()p Fy(s)p Fz(\)\))32 b(,)2105 4974 y Fp(Z)2204 5000 y Fs(e)2258 5110 y Fy(!)58 b Fz(=)2509 4974 y Fp(Z)2625 5110 y Fy(ds)2739 4974 y Fp(Z)2854 5110 y Fy(!)t Fz(\(0\)\()p Fy(s)p Fz(\))32 b(.)-24 5362 y(F)-8 b(or)32 b(this)h(construction,)h(w)m(e)g(assumed)g(that)e(\(\012)p Fy(;)17 b(d;)1920 5282 y Fp(R)1986 5362 y Fz(\))33 b(is)g(an)f Fy(\013)q Fz(-in)m(v)-5 b(arian)m(t)33 b Fy(n)p Fz(-cycle)h(for)e Fy(\021)t Fz(.)1935 5611 y(28)p eop end %%Page: 29 29 TeXDict begin 29 28 bop -24 171 a Fw(Prop)s(osition)38 b(9)49 b Fv(L)-5 b(et)39 b Fy(\021)k Fv(b)-5 b(e)39 b(an)g Fy(\013)q Fv(-invariant)g(cyclic)g(c)-5 b(o)g(cycle)38 b(on)h Ft(B)k Fv(and)c Fy(x)h Fv(b)-5 b(e)39 b(a)g(r)-5 b(epr)g(esentative)38 b(for)i(an)-24 291 y(element)35 b(in)f Fy(K)540 306 y Fs(i)568 291 y Fz(\()p Fy(S)6 b Ft(B)26 b Fu(o)840 306 y Fs(\034)8 b Fx(\012)p Fs(\013)1006 291 y Fu(R)p Fz(\))p Fv(.)44 b(Then)1334 511 y Ft(h)p Fz(#)1454 526 y Fs(\034)8 b Fx(\012)p Fs(\013)1597 511 y Fy(\021)1649 470 y Fs(s)1686 511 y Fy(;)17 b(x)p Ft(i)55 b Fz(=)g Ft(\000)2144 444 y Fz(1)p 2114 488 108 4 v 2114 579 a(2)p Fy(\031)2260 511 y Ft(h)p Fy(\021)2351 470 y Fs(e)2387 511 y Fy(;)17 b(x)p Ft(i)35 b Fv(.)1229 b Fz(\(25\))123 719 y(The)49 b(follo)m(wing)g(t)m(w)m(o)g(isomorphisms)h ([R82)o(])f(allo)m(w)f(to)g(reduce)i(the)f(pro)s(of)e(of)h(Prop)s (osition)g(9)g(to)g(a)-24 840 y(calculation)34 b(for)e(compact)h(op)s (erators:)595 1081 y(\011)28 b(:)g Fy(S)6 b Ft(B)25 b Fu(o)987 1096 y Fs(\034)8 b Fx(\012)p 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2438 y Fz(#)1532 2453 y Fs(\034)8 b Fx(\012)p Fs(\013)1692 2438 y Fy(\021)1744 2397 y Fs(s)1836 2438 y Fz(=)55 b(\011)2043 2397 y Fx(\003)2082 2438 y Fz(#)2163 2453 y Fs(\034)8 b Fx(\012)p Fl(id)2330 2438 y Fy(\021)2382 2397 y Fs(s)2453 2438 y Fv(,)1336 b Fz(\(28\))1542 2614 y Fy(\021)1594 2573 y Fs(e)1686 2614 y Fz(=)56 b(\011)1894 2573 y Fx(\003)1933 2614 y Fz(\010)2003 2573 y Fx(\003)2043 2614 y Fz(T)-8 b(r)22 b Ft(\012)h Fy(\021)38 b Fv(.)1438 b Fz(\(29\))-24 2792 y Fw(Pro)s(of:)33 b Fz(Let)g Fy(\021)j Fz(b)s(e)d(the)g(c)m(haracter)g(of)f(\(\012)p Fy(;)17 b(d;)1655 2711 y Fp(R)1721 2792 y Fz(\).)43 b(One)33 b(has)684 3021 y(#)765 3036 y Fs(\034)8 b Fx(\012)p Fs(\013)925 3021 y Fy(\021)977 2980 y Fs(s)1014 3021 y Fz(\()p Fy(f)1100 3036 y Fr(0)1139 3021 y Fy(;)17 b(:)g(:)g(:)f(;)h(f)1406 3036 y Fs(n)1453 3021 y Fz(\))55 b(=)1677 2885 y Fp(Z)1732 3111 y Fk(R)1801 3021 y Fy(ds)1915 2885 y Fp(Z)2031 2940 y(\000)2076 3021 y Fy(f)2124 3036 y Fr(0)2164 3021 y Fy(d)2215 2980 y Fs(s)2215 3046 y(\034)8 b Fx(\012)p Fs(\013)2358 3021 y Fy(f)2406 3036 y Fr(1)2462 3021 y Ft(\001)17 b(\001)g(\001)e Fy(d)2646 2980 y Fs(s)2646 3046 y(\034)8 b Fx(\012)p Fs(\013)2789 3021 y Fy(f)2837 3036 y Fs(n)2884 2940 y Fp(\001)2946 3021 y Fz(\(0\)\()p Fy(s)p Fz(\))32 b(,)-24 3244 y(where)i(the)f(pro)s(duct)g(is)g(that)f (in)h Fy(S)6 b Fz(\012)23 b Fu(o)1451 3259 y Fs(\034)8 b Fx(\012)p Fs(\013)1617 3244 y Fu(R)p Fz(.)43 b(On)33 b(the)g(other)f(hand)412 3465 y(\(\011)526 3423 y Fx(\003)565 3465 y Fz(#)646 3480 y Fs(\034)8 b Fx(\012)p Fl(id)812 3465 y Fy(\021)864 3423 y Fs(s)901 3465 y Fz(\)\()p Fy(f)1025 3480 y Fr(0)1064 3465 y Fy(;)17 b(:)g(:)g(:)f(;)h(f)1331 3480 y Fs(n)1378 3465 y Fz(\))27 b(=)1547 3329 y Fp(Z)1602 3555 y Fk(R)1671 3465 y Fy(ds)1785 3329 y Fp(Z)1900 3384 y(\000)1946 3465 y Fz(\011\()p Fy(f)2108 3480 y Fr(0)2147 3465 y Fz(\))p Fy(d)2236 3423 y Fs(s)2236 3489 y(\034)8 b Fx(\012)p Fl(id)2385 3465 y Fz(\011\()p Fy(f)2547 3480 y Fr(1)2587 3465 y Fz(\))17 b Ft(\001)g(\001)g(\001)d Fy(d)2825 3423 y Fs(s)2825 3489 y(\034)8 b Fx(\012)p Fl(id)2974 3465 y Fz(\011\()p Fy(f)3136 3480 y Fs(n)3183 3465 y Fz(\))3221 3384 y Fp(\001)3283 3465 y Fz(\(0\)\()p Fy(s)p Fz(\))p Fy(;)-24 3693 y Fz(with)34 b(pro)s(duct)e(in)h Fy(S)6 b Fz(\012)23 b Fu(o)914 3708 y Fs(\034)8 b Fx(\012)p Fl(id)1085 3693 y Fu(R)p Fz(.)44 b(No)m(w)33 b(\(28\))f(follo)m(ws)h (from)g Fy(d)2258 3657 y Fs(s)2258 3717 y(\034)8 b Fx(\012)p Fl(id)2406 3693 y Fz(\011\()p Fy(f)j Fz(\))28 b(=)f(\011\()p Fy(d)2913 3657 y Fs(s)2913 3717 y(\034)8 b Fx(\012)p Fs(\013)3056 3693 y Fy(f)j Fz(\))32 b(and)438 3788 y Fp(\000)483 3868 y Fz(\011\()p Fy(f)645 3883 y Fr(0)685 3868 y Fz(\))p Fy(d)774 3827 y Fs(s)774 3893 y(\034)8 b Fx(\012)p Fl(id)923 3868 y Fz(\011\()p Fy(f)1085 3883 y Fr(1)1124 3868 y Fz(\))17 b Ft(\001)g(\001)g(\001)d Fy(d)1362 3827 y Fs(s)1362 3893 y(\034)8 b Fx(\012)p Fl(id)1511 3868 y Fz(\011\()p Fy(f)1673 3883 y Fs(n)1720 3868 y Fz(\))1758 3788 y Fp(\001)1820 3868 y Fz(\(0\)\()p Fy(s)p Fz(\))55 b(=)g Fy(\013)2315 3883 y Fs(s)2369 3788 y Fp(\000)2414 3868 y Fy(f)2462 3883 y Fr(0)2502 3868 y Fy(d)2553 3827 y Fs(s)2553 3893 y(\034)8 b Fx(\012)p Fs(\013)2696 3868 y Fy(f)2744 3883 y Fr(1)2800 3868 y Ft(\001)17 b(\001)g(\001)e Fy(d)2984 3827 y Fs(s)2984 3893 y(\034)8 b Fx(\012)p Fs(\013)3127 3868 y Fy(f)3175 3883 y Fs(n)3222 3788 y Fp(\001)3284 3868 y Fz(\(0\)\()p Fy(s)p Fz(\))-24 4044 y(and)33 b(the)g Fy(\013)q Fz(-in)m(v)-5 b(ariance)33 b(of)f Fy(\021)t Fz(.)123 4164 y(F)-8 b(or)31 b(\(29\),)h(let)h Fy(f)719 4179 y Fs(j)784 4164 y Fz(=)27 b(\011)963 4128 y Fx(\000)p Fr(1)1057 4164 y Fz(\()p Fy(g)1142 4179 y Fs(j)1201 4164 y Ft(\012)22 b Fy(b)1341 4179 y Fs(j)1378 4164 y Fz(\))33 b(where)h Fy(g)1778 4179 y Fs(j)1836 4164 y Ft(\012)23 b Fy(b)1977 4179 y Fs(j)2041 4164 y Ft(2)28 b Fy(S)6 b Fu(C)22 b(o)2372 4179 y Fs(\034)2438 4164 y Fu(R)g Ft(\012)h(B)2728 4137 y(\030)2729 4168 y Fz(=)2833 4164 y Fy(S)6 b Ft(B)25 b Fu(o)3066 4179 y Fs(\034)8 b Fx(\012)p Fl(id)3238 4164 y Fu(R)p Fz(.)44 b(Then)233 4340 y(\011)309 4299 y Fx(\003)349 4340 y Fz(\010)419 4299 y Fx(\003)475 4340 y Fz(T)-8 b(r)23 b Ft(\012)f Fy(\021)t Fz(\()p Fy(f)835 4355 y Fr(0)874 4340 y Fy(;)17 b(:)g(:)g(:)f(;)h(f)1141 4355 y Fs(n)1188 4340 y Fz(\))83 b(=)g(T)-8 b(r)22 b Ft(\012)h Fy(\021)t Fz(\()p Fy(\032)p Fz(\()p Fy(g)1915 4355 y Fr(0)1954 4340 y Fz(\))f Ft(\012)h Fy(b)2155 4355 y Fr(0)2194 4340 y Fy(;)17 b(:)g(:)g(:)f(;)h(\032)p Fz(\()p Fy(g)2548 4355 y Fs(n)2595 4340 y Fz(\))22 b Ft(\012)h Fy(b)2796 4355 y Fs(n)2843 4340 y Fz(\))1309 4533 y(=)83 b(T)-8 b(r\()p Fy(\032)p Fz(\()p Fy(g)1741 4548 y Fr(0)1797 4533 y Ft(\001)17 b(\001)g(\001)e Fy(g)1977 4548 y Fs(n)2024 4533 y Fz(\)\))2117 4398 y Fp(Z)2232 4533 y Fy(b)2273 4548 y Fr(0)2313 4533 y Fy(db)2405 4548 y Fr(1)2461 4533 y Ft(\001)i(\001)g(\001)e Fy(db)2686 4548 y Fs(n)2761 4533 y Fy(;)660 4788 y(\021)712 4747 y Fs(e)749 4788 y Fz(\()p Fy(f)835 4803 y Fr(0)874 4788 y Fy(;)i(:)g(:)g(:)f(;)h(f)1141 4803 y Fs(n)1188 4788 y Fz(\))83 b(=)1468 4652 y Fp(Z)1523 4878 y Fk(R)1592 4652 y Fp(Z)1708 4788 y Fz(ev)1803 4803 y Fr(0)1865 4788 y Fz(\011)1941 4747 y Fx(\000)p Fr(1)2035 4788 y Fz(\()p Fy(g)2120 4803 y Fr(0)2181 4788 y Ft(\012)23 b Fy(b)2322 4803 y Fr(0)2361 4788 y Fz(\))p Fy(d)p Fz(\011)2526 4747 y Fx(\000)p Fr(1)2620 4788 y Fz(\()p Fy(g)2705 4803 y Fr(1)2767 4788 y Ft(\012)f Fy(b)2907 4803 y Fr(1)2947 4788 y Fz(\))17 b Ft(\001)g(\001)g(\001)d Fy(d)p Fz(\011)3261 4747 y Fx(\000)p Fr(1)3355 4788 y Fz(\()p Fy(g)3440 4803 y Fs(n)3509 4788 y Ft(\012)23 b Fy(b)3650 4803 y Fs(n)3697 4788 y Fz(\))1309 5047 y(=)1468 4911 y Fp(Z)1523 5137 y Fk(R)1592 5047 y Fz(ev)1687 5062 y Fr(0)1726 5047 y Fz(\()p Fy(g)1811 5062 y Fr(0)1867 5047 y Ft(\001)17 b(\001)g(\001)e Fy(g)2047 5062 y Fs(n)2093 5047 y Fz(\))2148 4911 y Fp(Z)2264 5047 y Fy(b)2305 5062 y Fr(0)2345 5047 y Fy(db)2437 5062 y Fr(1)2493 5047 y Ft(\001)i(\001)g(\001)e Fy(b)2667 5062 y Fs(n)2742 5047 y Fy(;)-24 5280 y Fz(and)33 b(the)g(lemma)g(follo)m(ws)h(from)e(the)h(fact)f(that)h(T)-8 b(r\()p Fy(\032)p Fz(\()p Fy(g)t Fz(\)\))27 b(=)2255 5200 y Fp(R)2302 5315 y Fk(R)2370 5280 y Fy(ds)17 b(g)t Fz(\(0\)\()p Fy(s)p Fz(\))31 b(for)h Fy(g)f Ft(2)d Fy(S)6 b Fu(C)22 b(o)3371 5295 y Fs(\034)3437 5280 y Fu(R)p Fz(.)382 b Fd(2)1935 5611 y Fz(29)p eop end %%Page: 30 30 TeXDict begin 30 29 bop -24 171 a Fw(Lemma)39 b(3)49 b Fv(L)-5 b(et)34 b Fy(\021)j Fv(b)-5 b(e)33 b(an)g Fy(\013)q Fv(-invariant)f(cyclic)h(c)-5 b(o)g(cycle)33 b(over)g Ft(B)s Fv(.)45 b(Then)32 b(for)i(a)f(pr)-5 b(oje)g(ction)32 b Fy(X)k Ft(2)28 b(K)20 b(\012)f(B)37 b Fv(or)-24 291 y(a)e(unitary)h Fy(X)f Ft(2)28 b Fy(U)10 b Fz(\()p Ft(K)24 b(\012)f(B)s Fz(\))35 b Fv(one)f(has)1122 482 y Ft(h)p Fz(T)-8 b(r)22 b Ft(\012)h Fy(\021)t(;)17 b(X)8 b Ft(i)54 b Fz(=)h Ft(\000)17 b Fz(2)p Fy(\031)32 b Ft(h)p Fz(#)2142 497 y Fs(\034)8 b Fx(\012)p Fl(id)2291 482 y Fy(\021)2343 441 y Fs(s)2380 482 y Fy(;)17 b Fz(\010)2494 441 y Fx(\000)p Fr(1)2588 482 y Fz(\()p Fy(X)8 b Fz(\))p Ft(i)27 b Fy(:)-24 676 y Fw(Pro)s(of:)38 b Fz(W)-8 b(e)37 b(start)h(with)g(the)f(case)i Ft(B)g Fz(=)d Fu(C)h Fz(and)g Fy(\021)j Fz(=)c(T)-8 b(r)37 b(whic)m(h)i(is)f(the)g(c)m(haracter)g(of)f(\()p Fu(C)p Fy(;)17 b Fz(0)p Fy(;)g Fz(T)-8 b(r)o(\).)58 b(Then)-24 797 y(T)-8 b(r)33 b Ft(\012)g Fy(\021)56 b Fz(=)d(T)-8 b(r)48 b(and)g(#)886 812 y Fs(\034)8 b Fx(\012)p Fl(id)1035 797 y Fy(\021)1087 761 y Fs(s)1177 797 y Fz(=)53 b(#)1387 812 y Fs(\034)1430 797 y Fz(T)-8 b(r)1530 755 y Fs(s)1614 797 y Fz(is)48 b(the)g(c)m(haracter)h(of)e(\()p Fy(S)6 b Fu(C)32 b(o)2762 812 y Fs(\034)2837 797 y Fu(R)h Ft(\012)g Fz(\003)p Fu(C)3192 761 y Fr(2)3231 797 y Fy(;)17 b(\016)n(;)3360 717 y Fp(R)3426 743 y Fs(s)3407 831 y(\034)3463 797 y Fz(\))47 b(where,)53 b(for)-24 917 y Fy(f)e Fz(:)40 b Fu(R)g Ft(!)g Fy(S)6 b Fu(C)p Fz(,)41 b Fy(\016)j Fz(=)c Fy(\016)846 932 y Fr(1)913 917 y Fz(+)27 b Fy(\016)1059 932 y Fr(2)1138 917 y Fz(with)41 b Fy(\016)1411 932 y Fr(1)1450 917 y Fz(\()p Fy(f)11 b Fz(\))40 b(=)g Fy(@)1792 932 y Fs(s)1829 917 y Fy(f)e Ft(\012)28 b Fy(e)2065 932 y Fr(1)2104 917 y Fz(,)42 b Fy(\016)2216 932 y Fr(2)2256 917 y Fz(\()p Fy(f)11 b Fz(\))39 b(=)h Ft(r)2629 932 y Fs(x)2673 917 y Fy(f)e Ft(\012)27 b Fy(e)2908 932 y Fr(2)2948 917 y Fz(,)42 b Ft(r)3100 932 y Fs(x)3144 917 y Fy(f)11 b Fz(\()p Fy(x)p Fz(\))40 b(=)g Fy({xf)11 b Fz(\()p Fy(x)p Fz(\),)42 b(and)-24 957 y Fp(R)43 984 y Fs(s)23 1072 y(\034)96 1038 y Fy(f)34 b Ft(\012)24 b Fy(e)324 1053 y Fr(1)364 1038 y Fy(e)409 1053 y Fr(2)479 1038 y Fz(=)586 957 y Fp(R)633 1072 y Fk(R)702 1038 y Fy(dsf)11 b Fz(\(0\)\()p Fy(s)p Fz(\))30 b(=)h(T)-8 b(r\()p Fy(\032)p Fz(\()p Fy(f)11 b Fz(\)\).)49 b(Let)35 b Fy(M)45 b Fz(and)34 b Fy(D)j Fz(b)s(e)e(op)s(erators)f(giv)m(en)i(b)m(y)f Fy(M)10 b( )t Fz(\()p Fy(x)p Fz(\))32 b(=)f Fy({x )t Fz(\()p Fy(x)p Fz(\))-24 1158 y(and)39 b Fy(D)s( )t Fz(\()p Fy(x)p Fz(\))f(=)g Fy( )673 1122 y Fx(0)697 1158 y Fz(\()p Fy(x)p Fz(\))h(with)g(usual)g(common)h(domain)f Fy(C)2183 1122 y Fr(1)2176 1183 y Fs(c)2222 1158 y Fz(\()p Fu(R)p Fz(\).)62 b(They)40 b(satisfy)g([)p Fy(D)s(;)17 b(M)10 b Fz(])38 b(=)g Fy({)p Fz(.)62 b(Then,)42 b(for)-24 1278 y(di\013eren)m(tiable)35 b Fy(f)11 b Fz(,)769 1469 y Ft(h)p Fy(x)p Ft(j)p Fz([)p Fy(M)5 b(;)17 b(\032)p Fz(\()p Fy(f)11 b Fz(\)])p Ft(j)p Fy(y)t Ft(i)54 b Fz(=)h Fy({)p Fz(\()p Fy(x)23 b Ft(\000)g Fy(y)t Fz(\))p Fy(f)11 b Fz(\()p Fy(x)21 b Ft(\000)h Fy(y)t Fz(\)\()p Fy(x)p Fz(\))55 b(=)g Ft(h)p Fy(x)p Ft(j)p Fy(\032)p Fz(\()p Ft(r)2886 1484 y Fs(x)2930 1469 y Fy(f)11 b Fz(\))p Ft(j)p Fy(y)t Ft(i)26 b Fy(;)772 1660 y Ft(h)p Fy(x)p Ft(j)p Fz([)p Fy(D)s(;)17 b(\032)p Fz(\()p Fy(f)11 b Fz(\)])p Ft(j)p Fy(y)t Ft(i)53 b Fz(=)i(\()p Fy(@)1653 1675 y Fs(x)1720 1660 y Fz(+)22 b Fy(@)1869 1675 y Fs(y)1911 1660 y Fz(\))p Fy(f)11 b Fz(\()p Fy(x)22 b Ft(\000)g Fy(y)t Fz(\)\()p Fy(x)p Fz(\))55 b(=)g Ft(h)p Fy(x)p Ft(j)p Fy(\032)p Fz(\()p Fy(@)2890 1675 y Fs(s)2928 1660 y Fy(f)11 b Fz(\))p Ft(j)p Fy(y)t Ft(i)26 b Fy(;)-24 1821 y Fz(and)33 b(therefore,)g(if)g Fy(p)27 b Ft(2)h Fy(S)6 b Fu(C)22 b(o)1099 1836 y Fs(\034)1165 1821 y Fu(R)33 b Fz(is)g(a)f(di\013eren)m(tiable)i(pro)5 b(jection,)563 2051 y Ft(h)p Fz(#)683 2066 y Fs(\034)727 2051 y Fz(T)-8 b(r)827 2009 y Fs(s)864 2051 y Fy(;)17 b(p)p Ft(i)54 b Fz(=)1237 1984 y(1)p 1192 2028 139 4 v 1192 2119 a(2)p Fy(\031)t({)1368 2051 y Fz(T)-8 b(r)17 b(\()p Fy(\032)p Fz(\()p Fy(p)p Fz(\)[[)p Fy(D)s(;)g(\032)p Fz(\()p Fy(p)p Fz(\)])p Fy(;)g Fz([)p Fy(M)5 b(;)17 b(\032)p Fz(\()p Fy(p)p Fz(\)]]\))55 b(=)h Ft(\000)2883 1984 y Fz(1)p 2854 2028 108 4 v 2854 2119 a(2)p Fy(\031)2999 2051 y Fz(T)-8 b(r\()p Fy(\032)p Fz(\()p Fy(p)p Fz(\)\))28 b Fy(:)414 b Fz(\(30\))-24 2277 y(In)24 b(the)g(last)f(equation,)j(w)m (e)f(used)f Fy(P)14 b Fz([[)p Fy(D)s(;)j(P)d Fz(])p Fy(;)j Fz([)p Fy(M)5 b(;)17 b(P)d Fz(]])p Fy(P)39 b Fz(=)28 b Ft(\000)p Fy(P)14 b Fz([)p Fy(D)s(;)j(M)10 b Fz(])p Fy(P)17 b Fz(+)s([)p Fy(P)d(D)s(P)s(;)j(P)d(M)c(P)k Fz(],)23 b Fy(P)41 b Fz(=)28 b Fy(\032)p Fz(\()p Fy(p)p Fz(\))g(=)f Fy(P)3926 2241 y Fr(2)3965 2277 y Fz(.)-24 2397 y(This)k(pro)m(v)m(es)h (the)e(statemen)m(t)h(for)e Ft(B)j Fz(=)27 b Fu(C)j Fz(and)g Fy(\021)h Fz(=)d(T)-8 b(r.)42 b(Note)30 b(that)g(this)g(calculation)h (is)f(the)g(usual)g(one)g(to)-24 2518 y(obtain)h(the)g(Chern)h(n)m(um)m (b)s(er)g(of)e(the)i(Landau)e(pro)5 b(jection)32 b(\()p Fy(\015)g Fz(=)c(1\))i(in)h(the)h(Sc)m(hr\177)-49 b(odinger)32 b(represen)m(tation)g(.)123 2685 y(In)g(the)g(general)h(case,)g Fy(\021)i Fz(is)e(the)f(c)m(haracter)h(of)f(some)g Fy(n)p Fz(-cycle)i(\(\012)p Fy(;)17 b(d;)2686 2605 y Fp(R)2752 2685 y Fz(\))32 b(o)m(v)m(er)h Ft(B)s Fz(.)43 b(Then)33 b Fy(d)3473 2649 y Fs(s)3473 2710 y(\034)8 b Fx(\012)p Fl(id)3650 2685 y Fz(=)28 b Fy(d)3805 2649 y Fx(0)3848 2685 y Fz(+)21 b Fy(\016)-24 2806 y Fz(where)34 b Fy(\016)j Fz(is)c(as)g(ab)s(o)m(v)m(e)g(and)g Fy(d)1073 2770 y Fx(0)1096 2806 y Fz(\()p Fy(f)11 b Fz(\)\()p Fy(x)p Fz(\)\()p Fy(s)p Fz(\))27 b(=)g Fy(d)p Fz(\()p Fy(f)11 b Fz(\()p Fy(x)p Fz(\)\()p Fy(s)p Fz(\)\).)123 2973 y(W)-8 b(e)34 b(apply)i(this)f(\014rst)f(to)g(a)h(pro)5 b(jection)35 b(of)f(the)g(form)h Fy(X)j Fz(=)31 b Fy(\032)p Fz(\()p Fy(p)p Fz(\))23 b Ft(\012)h Fy(x)31 b Ft(2)h(K)24 b(\012)g(B)3120 2937 y Fr(+)3214 2973 y Fz(where)36 b Fy(x)e Fz(and)h Fy(p)f Fz(are)-24 3094 y(pro)5 b(jections.)51 b(Then)36 b Fy(d)830 3058 y Fs(s)830 3118 y(\034)8 b Fx(\012)p Fl(id)978 3094 y Fz(\(\010)1086 3058 y Fx(\000)p Fr(1)1181 3094 y Fz(\()p Fy(X)g Fz(\)\))31 b(=)g Fy(p)23 b Ft(\012)h Fy(dx)g Fz(+)f Fy(\016)t(p)g Ft(\012)h Fy(x)p Fz(.)50 b(Let)34 b Fy(n)e Fz(=)f(2)p Fy(k)s Fz(,)j Fy(k)g Ft(\025)e Fz(0.)49 b(Using)35 b Fy(p)p Fz(\()p Fy(\016)t(p)p Fz(\))p Fy(p)3678 3058 y Fs(j)3745 3094 y Fz(=)c(0)j(if)-24 3214 y Fy(j)g(>)28 b Fz(0,)k Fy(x)p Fz(\()p Fy(dx)p Fz(\))499 3178 y Fs(l)q Fx(\000)p Fr(1)616 3214 y Fy(x)c Fz(=)f(0)32 b(for)h Fy(l)c(>)f Fz(1)k(and)1466 3134 y Fp(R)1513 3249 y Fk(R)1582 3214 y Fy(ds)17 b Fz(ev)1790 3229 y Fr(0)1829 3214 y Fz(\()p Fy(p)p Fz(\()p Fy(\016)t(p)p Fz(\))2088 3178 y Fr(2)2127 3214 y Fz(\))28 b(=)g(2)p Fy(\031)t({)f Ft(h)p Fz(#)2583 3229 y Fs(\034)2626 3214 y Fz(T)-8 b(r)2726 3172 y Fs(s)2763 3214 y Fy(;)17 b(p)p Ft(i)27 b Fz(=)h Ft(\000)17 b Fy({)28 b Fz(T)-8 b(r\()p Fy(\032)p Fz(\()p Fy(p)p Fz(\)\),)33 b(w)m(e)g(get)306 3465 y Ft(h)p Fz(#)426 3480 y Fs(\034)8 b Fx(\012)p Fl(id)592 3465 y Fy(\021)644 3424 y Fs(s)681 3465 y Fy(;)17 b Fz(\010)795 3424 y Fx(\000)p Fr(1)889 3465 y Fz(\()p Fy(X)8 b Fz(\))p Ft(i)55 b Fz(=)g Fy(c)1321 3480 y Fr(2)p Fs(k)r Fr(+2)1506 3330 y Fp(Z)1605 3356 y Fs(s)1561 3555 y(\034)8 b Fx(\012)p Fl(id)1727 3465 y Fz(\010)1797 3424 y Fx(\000)p Fr(1)1892 3465 y Fz(\()p Fy(X)g Fz(\()p Fy(d)2108 3424 y Fs(s)2108 3490 y(\034)g Fx(\012)p Fl(id)2256 3465 y Fy(X)g Fz(\))2383 3424 y Fr(2)p Fs(k)r Fr(+2)2551 3465 y Fz(\))389 3731 y(=)56 b Fy(c)563 3746 y Fr(2)p Fs(k)r Fr(+2)884 3636 y Fp(X)747 3848 y Fr(0)p Fs()e Fz(0,)i(one)g(obtains,)h(taking)f(in)m(to)-24 4765 y(accoun)m(t)d(that) e Fy(p)p Fz(\()p Fy(\016)t(p)p Fz(\))p Fy(p)818 4728 y Fs(k)888 4765 y Fz(=)c(0)k(for)g Fy(k)f(>)c Fz(0,)13 5006 y Ft(h)p Fz(#)133 5021 y Fs(\034)8 b Fx(\012)p Fl(id)299 5006 y Fy(\021)351 4965 y Fs(s)387 5006 y Fy(;)17 b Fz(\010)501 4965 y Fx(\000)p Fr(1)596 5006 y Fz(\()p Fy(X)8 b Fz(\))o Ft(i)55 b Fz(=)h Fy(c)1028 5021 y Fr(2)p Fs(k)r Fr(+3)1212 4870 y Fp(Z)1312 4897 y Fs(s)1268 5096 y(\034)8 b Fx(\012)p Fl(id)1433 5006 y Fz(\010)1503 4965 y Fx(\000)p Fr(1)1598 5006 y Fz(\(\()p Fy(X)1763 4965 y Fx(\003)1824 5006 y Ft(\000)23 b Fz(1\))p Fy(d)2062 4965 y Fs(s)2062 5031 y(\034)8 b Fx(\012)p Fl(id)2211 5006 y Fy(X)g Fz(\()p Fy(d)2389 4965 y Fs(s)2389 5031 y(\034)g Fx(\012)p Fl(id)2537 5006 y Fy(X)2626 4965 y Fx(\003)2666 5006 y Fy(d)2717 4965 y Fs(s)2717 5031 y(\034)g Fx(\012)p Fl(id)2866 5006 y Fy(X)g Fz(\))2993 4965 y Fs(k)r Fr(+1)3125 5006 y Fz(\))13 5272 y(=)27 b Fy(c)158 5287 y Fr(2)p Fs(k)r Fr(+3)343 5136 y Fp(Z)398 5362 y Fk(R)467 5272 y Fy(ds)17 b Fz(ev)675 5287 y Fr(0)731 5191 y Fp(\000)777 5272 y Fy(p)p Fz(\()p Fy(\016)t(p)p Fz(\))998 5231 y Fr(2)1037 5191 y Fp(\001)1198 5177 y(X)1099 5389 y Fr(0)p Fs(