This is a multi-part message in MIME format. ---------------0411220746714 Content-Type: text/plain; name="04-396.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-396.keywords" Magnetic field, pseudodifferential operator, Weyl calculus, strict deformation quantization, Moyal product, twisted crossed product, Rieffel's axioms. ---------------0411220746714 Content-Type: application/postscript; name="mp-rieffel-6.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="mp-rieffel-6.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.92b Copyright 2002 Radical Eye Software %%Title: mp-rieffel-6.dvi %%Pages: 13 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMR17 CMR12 CMSY8 CMBX9 CMR9 CMMI9 CMSY6 MSBM10 CMSY9 %%+ CMBX10 CMR10 CMBX12 CMMI10 CMSY7 CMTI10 EUFM10 MSBM7 CMSY10 CMMI7 %%+ CMR7 CMR8 CMEX10 CMMI5 CMSY5 CMMI12 MSBM5 CMR5 MSAM10 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: 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All Rights Reserved) readonly def /FullName (MSAM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSAM10 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 4 /squaresolid put readonly def /FontBBox{8 -463 1331 1003}readonly def /UniqueID 5031981 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF7158F1163BC1C87678CE98C24B934A76220 4DD9B2FF3A49786028E35DDE10AD2C926BD30AD47015FFE9469DE1F793D1C53A C8812CBCD402444EAEA7A50EC5FD93D6A04C2783B50EA48059E3E7407537CB8D 4C206846EF0764C05289733920E2399E58AD8F137C229F3CE3E34D2D1EAB2D53 20D44EFAC8EFA4D14A2EFE389D952527F98D0E49BD5BD2C8D58FF9CB9C78D974 75C2AB5467D73D2B5E277A3FDC35909938A9DF0EB91BD9159D3437BE22EE4544 3429AC8E2BFBE34AE54D3BA3AD04BDF3F4F43A2B43992DF88678681B3AB32CFD A23E2C98D1AF00AB206AC95B78BBE6316F7A0AB6BD3236C28C76288B3C25D1EB E9ABB3576C5EC15A71D26177F5883E9B48293D59015615E2EEAF2E9BA04151ED 5497B9A1C41CBA44BAFF13EA218F5EAC11952EE336AD1DBE6CE92F002EAA3B3D 3BE4C3792F3405763C4BD93EFC3B4FC34193439561841BA989DD8D9F9AEE7A7B 24AEB4654B35023C9720B8F31AA9452E29753FB7915CB29977E725611E37C0B7 784BCC26FACF8A7A0EB1E54290D27FFE52B2D87FAD080AD15EE1984C37E0EB30 122C3012D3A16B09C28903D138352AB5462674B6CFB63F1371768D094DDF288C 36FB9B58443F872D61F2CD8CED42FE0EFF3D7E9952A172BB1AFECB60BF79F2B6 04265FDE4F78BC9FD619AA733CD0412F1D9A7C13B271BF827DCBDC8ABAE24FF0 74D3C220621D7FF0EFE62D835A221D0A7C139E2E6681FC2BBA58FA3B80D416EC 3854C63BA040A4262B458340DAA18AA6AEA3BBAC61615CB85982B18664D3D3AF 340C65B969071CF2D0CABEB80E04623D0526F862ECA8280EEE236C535F70561A 854181132E677674AD5E14C6636F57541D3C821F0776D2CB9B8526D4B826791A 0B179B387D47B08982D338AE4D93711A9AFEC6B628B8EC464346666841657E34 0A8624A495F142A3D4BEDF75EA6AEB3F283B1B527748949E4767368AB7391F22 80B8DD8461C76DABE568B977E3CBE4B17F7A5D58CB025CAB581588CFE46CE7F8 7D54E7688CE550D5F283BAA71FECD03B45052F6C5467A3A93B1B1AF15B611A31 C49998ECB8988D5CD38B4DFB9E7F4B5A363E2D703396B0DE5934CE210331B2DC 5D0A41A7321E0B76A1DC8D3129CA0FE67228C0D9B4A56B3D6551D1B34ADBB2B9 19B3C9FAA8B33B7B1FAABE696F303F11E2A1039860905E19 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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: MSBM5 %!PS-AdobeFont-1.1: MSBM5 2.1 %%CreationDate: 1992 Oct 17 08:36:07 % 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 (MSBM5) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM5 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 126 /planckover2pi1 put readonly def /FontBBox{0 -574 3027 1074}readonly def /UniqueID 5032013 def currentdict end currentfile eexec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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 /.notdef put readonly def /FontBBox{-34 -251 988 750}readonly def /UniqueID 5000794 def currentdict end currentfile eexec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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. 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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 (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 0 /.notdef put readonly def /FontBBox{-30 -250 1026 750}readonly def /UniqueID 5087386 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY5 %!PS-AdobeFont-1.1: CMSY5 1.0 %%CreationDate: 1991 Aug 15 07:21:16 % 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 (CMSY5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY5 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{21 -944 1448 791}readonly def /UniqueID 5000815 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI5 %!PS-AdobeFont-1.1: CMMI5 1.100 %%CreationDate: 1996 Aug 02 08:21:10 % 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 (CMMI5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI5 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{37 -250 1349 750}readonly def /UniqueID 5087380 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA06DA87FC7163A5A2A756A598FAB07633 89DE8BB201D5DB4627484A80A431B6AFDBBBF23D4157D4AFE17E6B1C853DD417 25F84CD55402AB88AB7EEFDEDBF2C2C731BD25567C53B474CCF739188A930039 098A197F9C4BE7594D79442B2C8A67447DE44698321145D7689B91EF235EA80E B600AA8E238064F154284096C4C2554EFE8DDF13AFF8D3CE30E0999375C0FEE6 F992DEA5FC3897E2CC8B7A90238E61E41622DE80F438DD994C73275CC52249D9 F6686F87F394FB7BB668138B210BEC9E46415A1B58C990B81E7D7DD301143517 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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 2 /bracketleftbig put dup 3 /bracketrightbig put dup 12 /vextendsingle put dup 13 /vextenddouble 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 68 /angbracketleftBig put dup 69 /angbracketrightBig put dup 80 /summationtext put dup 82 /integraltext put dup 88 /summationdisplay put dup 90 /integraldisplay put dup 104 /bracketleftBig put dup 105 /bracketrightBig put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueID 5000774 def currentdict end 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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 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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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 82 /R put dup 126 /planckover2pi1 put readonly def /FontBBox{0 -504 2615 1004}readonly def /UniqueID 5032014 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: EUFM10 %!PS-AdobeFont-1.1: EUFM10 2.1 %%CreationDate: 1992 Nov 20 17:36:20 % 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 (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 65 /A put dup 66 /B put dup 67 /C put dup 70 /F put dup 76 /L put dup 79 /O put dup 82 /R put dup 101 /e put dup 112 /p put readonly def /FontBBox{-26 -224 1055 741}readonly def /UniqueID 5031986 def currentdict end currentfile eexec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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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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. 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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. 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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. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY9 %!PS-AdobeFont-1.1: CMSY9 1.0 %%CreationDate: 1991 Aug 15 07:22:27 % 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 (CMSY9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY9 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 -958 1146 777}readonly def /UniqueID 5000819 def currentdict end currentfile eexec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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: CMMI9 %!PS-AdobeFont-1.1: CMMI9 1.100 %%CreationDate: 1996 Jul 23 07:53:55 % 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 (CMMI9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI9 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{-29 -250 1075 750}readonly def /UniqueID 5087384 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D5993DFC0930297866E1CD0A319B6B1FD958 9E394A533A081C36D6F5CA5FED4F9AC9ADE41E04F9FC52E758C9F45A92BED935 86F9CFDB57732045913A6422AD4206418610C81D882EE493DE9523CC1BFE1505 DD1390B19BC1947A01B93BC668BE9B2A0E69A968554239B88C00AF9FBDF09CCD 67D3B2094C11A04762FE8CC1E91D020A28B3C122D24BEAACF82313F4604F2FEF 6E176D730A879BE45DD0D4996EF0247AEB1CA0AB08FF374D99F06D47B36F9554 FAD9A2D3CE451B7791C3709D8A1DDDEFBD840C1B42AB824D5A0DFF0E0F15B0B7 22AEEB877FF489581DA6FA8DA64944555101EB16F7AB0B717E148B7B98D8DBFD 730C52937E226545CF8DC3E07C5BA30739BAFCD0F2B44275A6D503F582C0FB4F 449963D0AD2FAFDE33BA3D77BCA9D1DF878DDAFCA2E22CC4BACD542B282164C7 97C2BDE318AF9D501CA21F6E662E7AAB75A5F24D2C182E598D175D44E88AB19A E7CD59584F95B389183EE21B525BF52A3F23C0FE5383A5565A19361D716F508C AAB78411CA5A4D27552CC1C435760D5A89D535B71C593E755C616661363308DA A683F54ED0C23FB2C225A008392B0B719F66F11A946A090B7C00B662A3C69599 B4ECB0CC70C85C4BBBF207E0026F6C7A19F2ACFB7A60804FC98A4BFFD7BFFF2B 9529E6D9D4238002BBC255BC62959D6F3381FE06E0621B879D5FE5B541D45A1E 759A6E7DC32B1D1632368D09A97039DF255B6492B1B2B7E2C1434E8306ECA7D3 5A79B6D614B4979F10988BC76ED53A5F45315CD7DA216221F842FD0F3E050DD2 BAC23C984D506D8F7D614BCB6B244F5F41321549BB0BD041FBF3053307168680 3435E9C9445A59A7C666418C4F2512C32058B1CE1EA45F8184B6A5A77CB2CF55 24A461B677583E11B7ED2E542F116AC9F650280289310DEB441B047635CE26E9 0D3D0CAADAE852F21CDC4059272B0BEA1CF5595BD6EA1CB064D13E4C0B93163A ADE562E2A425E4E95033BE0A9DE377A4786EE6653C02B091652290AB0A210F13 5FD8DD83747FAFF007FB86DFA34FE11ACF825FF93FE2DD12FA27E5AD20DEAC38 D5158877F52F3289EBED83AE97F813EB852FAD1CEC3DFB6E8DF31B31A1D33749 55B631AFD60AE0BA7A2706486F74D7F53B7CFDA20ECA0B0E82EC5DD96AD60870 721196452F479300E3340384F0D96F605D8F8FA621789AC401F8BB84DE22F558 6F894175289C9233161A37B313B43F08B47F2D06D80F7D1A7C3D3435956727A0 6DF8BA48E9AACB96F76F2F91CEED42AC4E18CAC3994600FC6A16C6625FF75CC4 8296D71E589B51E35D5B20B5A15550D50096E8 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR9 %!PS-AdobeFont-1.1: CMR9 1.0 %%CreationDate: 1991 Aug 20 16:39:59 % 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 (CMR9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR9 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{-39 -250 1036 750}readonly def /UniqueID 5000792 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF7158F1163BC1F3352E22A1452E73FECA8A4 87100FB1FFC4C8AF409B2067537220E605DA0852CA49839E1386AF9D7A1A455F 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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 /.notdef put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueID 5000818 def currentdict end currentfile eexec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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 0 /.notdef put readonly def /FontBBox{-33 -250 945 749}readonly def /UniqueID 5000795 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5F075EA0A10A15B0ED05D5039DA41B32B16E95 A3CE9725A429B35BAD796912FC328E3A28F96FCADA20A598E247755E7E7FF801 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y(uniformly)k(con)n(tin)n(uous")e(and)i(\\con)n(tin)n(uous)e (and)i(small)g(at)g(in\014nit)n(y".)37 b(If)29 b Fn(H)g Fv(is)f(a)f(Hilb)r(ert)i(space,)e Ft(K)6 b Fv(\()p Fn(H)q Fv(\))28 b(will)g(b)r(e)g(the)h(set)f(of)-118 2047 y(all)j(compact)g (op)r(erators)e(in)j Fn(H)q Fv(,)g(forming)f(an)g(ideal)g(in)g(the)h Ft(C)1879 2017 y Fs(\003)1917 2047 y Fv(-algebra)d Ft(B)t Fv(\()p Fn(H)q Fv(\))j(of)g(all)f(the)g(linear,)h(b)r(ounded)g(op)r (erators)d(on)-118 2147 y Fn(H)q Fv(.)37 b(The)27 b(unitary)h(op)r (erators)d(form)i(the)h(group)f Fn(U)8 b Fv(\()p Fn(H)q Fv(\).)-118 2246 y Fw(Ac)m(kno)m(wledgemen)m(ts:)42 b Fv(A)30 b(large)f(part)h(of)h(this)f(w)n(ork)f(has)h(b)r(een)h (completed)g(while)f(the)h(authors)e(visited)i(the)g(Univ)n(ersit)n(y) -118 2346 y(of)g(Genev)-5 b(a)32 b(and)f(w)n(e)g(express)f(our)h (gratitude)g(to)g(Prof.)48 b(W)-7 b(erner)30 b(Amrein)i(for)f(his)g (kind)h(hospitalit)n(y)f(and)g(the)h(stim)n(ulating)-118 2445 y(discusions.)j(W)-7 b(e)26 b(also)f(ac)n(kno)n(wledge)e(the)j (partial)f(supp)r(ort)g(from)g(the)h(EURR)n(OMMA)-7 b(T)26 b(Programme)d(\(con)n(tract)h(no.)36 b(ICA1-)-118 2545 y(CT-2000-70022\))20 b(and)25 b(from)f(the)h(CERES)g(Programme)d(\(con) n(tract)i(no.)36 b(3-28/2003\).)c(W)-7 b(e)25 b(are)f(greatful)g(to)h (F)-7 b(r)n(\023)-39 b(ed)n(\023)g(eric)22 b(Cadet)-118 2645 y(for)27 b(a)g(useful)h(discussion)f(and)g(to)h(Serge)e(Ric)n (hard)h(for)g(a)g(critical)g(reading)g(of)g(the)h(man)n(uscript.)-118 2919 y Fu(1)135 b(The)44 b(axioms)-118 3101 y Fv(W)-7 b(e)24 b(describ)r(e)f(here)h(Rie\013el's)g(framew)n(ork)e(for)h (strict)g(quan)n(tization.)35 b(There)23 b(are)g(sev)n(eral)f(v)n (ersions)g(of)i(his)g(axioms;)g(w)n(e)f(c)n(ho)r(ose)-118 3201 y(to)33 b(w)n(ork)e(with)j(the)f(system)g(of)g(axioms)f(whic)n(h)h (app)r(ears)f(in)h([7],)h(to)f(whic)n(h)g(w)n(e)g(also)f(refer)g(for)h (man)n(y)f(other)h(details.)53 b(The)-118 3300 y(starting)27 b(p)r(oin)n(t)g(is)h(a)f(\\classical)f(algebra)g(of)h(observ)-5 b(ables")26 b(describ)r(ed)h(b)n(y)g(a)h(P)n(oisson)d(algebra.)-118 3483 y Fw(De\014nition)31 b(1.1)40 b Fv(A)29 b(P)n(oisson)d(algebra)i Fr(is)j(a)f(triple)i Fv(\()p Fq(A)p Ft(;)14 b Fn(\016)p Ft(;)g Fn(f\001)p Ft(;)g Fn(\001g)p Fv(\))n Fr(,)31 b(wher)l(e)g Fq(A)f Fr(is)g(a)h(r)l(e)l(al)f(ve)l(ctor)h(sp)l(ac)l(e,)62 b Fn(\016)p Ft(;)43 b Fn(f\001)p Ft(;)14 b Fn(\001g)30 b Fr(ar)l(e)g(biline)l(ar)-118 3583 y(maps)35 b Fv(:)d Fq(A)21 b Fn(\002)h Fq(A)31 b Fn(!)h Fq(A)i Fr(such)g(that)h Fn(\016)f Fr(is)g(asso)l(ciative)j(and)e(c)l(ommutative,)h Fn(f\001)p Ft(;)14 b Fn(\001g)34 b Fr(is)g(antisymmetric)h(and)g(for)h (e)l(ach)70 b Ft(')32 b Fn(2)g Fq(A)p Fr(,)-118 3682 y Fn(f)p Ft(';)14 b Fn(\001g)33 b Fr(is)h(a)g(derivation)i(b)l(oth)e (with)g(r)l(esp)l(e)l(ct)g(to)g Fn(\016)f Fr(and)h(to)g Fn(f\001)p Ft(;)14 b Fn(\001g)p Fr(.)50 b(Thus,)35 b(aside)h(biline)l (arity,)h(the)d(two)g(maps)g(satisfy)h(for)g(al)t(l)-118 3782 y Ft(';)14 b( )s(;)g(\032)23 b Fn(2)g Fq(A)p Fr(:)-118 3881 y(\(i\))30 b Ft( )21 b Fn(\016)d Ft(')24 b Fv(=)e Ft(')d Fn(\016)f Ft( )s(;)119 b Fv(\()p Ft( )22 b Fn(\016)c Ft(')p Fv(\))h Fn(\016)f Ft(\032)23 b Fv(=)f Ft( )g Fn(\016)c Fv(\()p Ft(')h Fn(\016)f Ft(\032)p Fv(\))p Fr(,)-118 3981 y(\(ii\))30 b Fn(f)p Ft( )s(;)14 b(')p Fn(g)23 b Fv(=)f Fn(\000f)p Ft(';)14 b( )s Fn(g)p Fr(,)-118 4081 y(\(iii\))31 b Fn(f)p Ft(';)14 b( )21 b Fn(\016)d Ft(\032)p Fn(g)23 b Fv(=)f Ft( )g Fn(\016)c(f)p Ft(';)c(\032)p Fn(g)j Fv(+)h Fn(f)p Ft(';)c( )s Fn(g)k(\016)g Ft(\032)89 b Fv(\(Leibnitz)28 b(rule\))p Fr(,)-118 4180 y(\(iv\))i Fn(f)p Ft(';)14 b Fn(f)p Ft( )s(;)g(\032)p Fn(gg)21 b Fv(=)i Fn(ff)p Ft(';)14 b( )s Fn(g)p Ft(;)g(\032)p Fn(g)j Fv(+)h Fn(f)p Ft( )s(;)c Fn(f)p Ft(';)g(\032)p Fn(gg)57 b Fv(\(Jacobi's)27 b(iden)n(tit)n(y\))p Fr(.)-118 4363 y Fv(The)f(elemen)n(ts)g(of)f Fq(A)h Fv(are)f(in)n(terpreted)g(as)g (observ)-5 b(ables)25 b(of)g(a)h(classical)e(description)i(of)f(a)h(ph) n(ysical)f(system.)36 b(F)-7 b(or)25 b(eac)n(h)g Ft(')f Fn(2)f Fq(A)-118 4463 y Fv(and)37 b(eac)n(h)f(v)-5 b(alue)37 b Fp(~)h Fn(6)p Fv(=)h(0)d(of)h(Planc)n(k's)e(constan)n(t,)k(one)e(w)n (ould)f(lik)n(e)h(to)g(ha)n(v)n(e)e(an)i(ob)5 b(ject)37 b Fn(Q)2890 4432 y Fo(~)2931 4463 y Fv(\()p Ft(')p Fv(\))g(represen)n (ting)f(the)h(same)-118 4562 y(observ)-5 b(able)24 b(in)h(a)g(quan)n (tum)g(description)f(of)h(the)h(system.)36 b(One)25 b(also)f(hop)r(es)g (that)i(the)f(algebraic)f(structure)g(of)h(the)h(quan)n(tum)-118 4662 y(observ)-5 b(ables)32 b(should)h(con)n(v)n(erge)e(to)i(the)h (classical)e(picture)h(describ)r(ed)g(b)n(y)g(the)h(P)n(oisson)e (algebra,)h(in)g(some)g(suitable)g(norm)-118 4761 y Fn(k)22 b(\001)i(k)35 4773 y Fo(~)98 4761 y Fv(dep)r(ending)f(con)n(tin)n (uously)e(of)i Fp(~)p Fv(.)35 b(This)22 b(migh)n(t)h(b)r(e)g(seen)f(as) g(a)g(precise)g(mathematical)g(form)h(of)f(Bohr's)g(corresp)r(ondence) -118 4861 y(principle.)-118 4961 y(A)33 b(systematic)f(justi\014cation) h(of)f(the)h(next)g(de\014nitions)g(ma)n(y)e(b)r(e)i(found)g(in)g([7].) 52 b(Note)32 b(that)h(usually)f(in)h Fq(A)f Fv(man)n(y)g(classical)-118 5060 y(observ)-5 b(ables)28 b(are)h(\\un)n(b)r(ounded";)i(the)f(use)g (of)g(norms)f(forces)g(us)h(to)g(apply)g(quan)n(tization)f(only)h(to)f (certain)h(subfamilies)g Fq(A)3861 5072 y Fl(0)-118 5160 y Fv(of)d Fq(A)p Fv(.)-118 5343 y Fw(De\014nition)k(1.2)40 b Fr(L)l(et)27 b Fq(A)692 5355 y Fl(0)755 5343 y Fr(b)l(e)g(a)g (Poisson)h(algebr)l(a)g(which)h(is)e(densely)g(c)l(ontaine)l(d)h(in)f (the)f(self-adjoint)j(p)l(art)e Fq(C)3364 5312 y Fl(0)3364 5366 y Fo(R)3437 5343 y Fr(of)g(an)g(ab)l(elian)-118 5442 y Ft(C)-53 5412 y Fs(\003)-15 5442 y Fr(-algebr)l(a)k Fq(C)347 5412 y Fl(0)413 5442 y Fr(\(with)f(the)g(same)g(asso)l (ciative)h(pr)l(o)l(duct\).)39 b Fv(A)27 b(strict)g(quan)n(tization)g (of)g(the)g(P)n(oisson)f(algebra)h(\()p Fq(A)3458 5454 y Fl(0)3495 5442 y Ft(;)14 b Fn(\016)p Ft(;)g Fn(f\001)p Ft(;)g Fn(\001g)p Fv(\))28 b Fr(is)-118 5542 y(a)i(family)i(of)e(maps) 521 5475 y Fj(\000)559 5542 y Fn(Q)627 5512 y Fo(~)690 5542 y Fv(:)23 b Fq(A)796 5554 y Fl(0)856 5542 y Fn(!)g Fq(C)1013 5512 y Fo(~)1013 5565 y(R)1059 5475 y Fj(\001)1097 5575 y Fo(~)p Fs(2)p Fm(I)1217 5542 y Fr(,)30 b(wher)l(e)-118 5650 y(\(i\))g Ft(I)37 b Fr(is)30 b(a)g(subset)f(of)i(the)f(r)l(e)l(al) g(axis,)g(for)h(which)g(the)f(origin)h(is)f(an)g(ac)l(cumulation)g(p)l (oint)g(c)l(ontaine)l(d)g(in)g Ft(I)7 b Fr(,)1869 5989 y Fv(2)p eop end %%Page: 3 3 TeXDict begin 3 2 bop -118 154 a Fr(\(ii\))44 b Fq(C)95 124 y Fo(~)178 154 y Fr(is)g(a)f Ft(C)431 124 y Fs(\003)470 154 y Fr(-algebr)l(a,)k(with)d(pr)l(o)l(duct)f(and)h(norm)f(denote)l(d) g(r)l(esp)l(e)l(ctively)h(by)g Ft(])2650 124 y Fo(~)2734 154 y Fr(and)f Fn(k)k(\001)g(k)3109 166 y Fo(~)3149 154 y Fr(.)79 b(F)-6 b(or)43 b Ft(')3475 124 y Fo(~)3516 154 y Ft(;)14 b( )3610 124 y Fo(~)3698 154 y Fn(2)48 b Fq(C)3852 124 y Fo(~)3852 177 y(R)-118 254 y Fr(\(the)37 b(self-adjoint)h(p)l(art)f(of)g Fq(C)833 223 y Fo(~)874 254 y Fr(\))f(we)h(set)g Ft(')1265 223 y Fo(~)1329 254 y Ft(?)1371 223 y Fo(~)1435 254 y Ft( )1492 223 y Fo(~)1568 254 y Fv(:=)1701 221 y Fl(1)p 1701 235 34 4 v 1701 282 a(2)1758 186 y Fj(\000)1796 254 y Ft(')1850 223 y Fo(~)1892 254 y Ft(])1924 223 y Fo(~)1965 254 y Ft( )2022 223 y Fo(~)2081 254 y Fv(+)18 b Ft( )2221 223 y Fo(~)2262 254 y Ft(])2294 223 y Fo(~)2336 254 y Ft(')2390 223 y Fo(~)2431 186 y Fj(\001)2505 254 y Fv(\(a)35 b(Jordan)f(pro)r(duct\))i Fr(and)i Fn(f)p Ft(')3517 223 y Fo(~)3557 254 y Ft(;)14 b( )3651 223 y Fo(~)3692 254 y Fn(g)3734 266 y Fo(~)3810 254 y Fv(:=)-95 329 y Fl(1)p -108 343 61 4 v -108 390 a Fm(i)p Fo(~)-24 294 y Fj(\000)14 362 y Ft(')68 331 y Fo(~)109 362 y Ft(])141 331 y Fo(~)182 362 y Ft( )239 331 y Fo(~)299 362 y Fn(\000)k Ft( )439 331 y Fo(~)480 362 y Ft(])512 331 y Fo(~)553 362 y Ft(')607 331 y Fo(~)648 294 y Fj(\001)686 362 y Fr(.)-118 461 y(\(iii\))31 b Fn(Q)124 431 y Fo(~)188 461 y Fv(:)23 b Fq(A)294 473 y Fl(0)354 461 y Fn(!)g Fq(C)511 431 y Fo(~)511 484 y(R)586 461 y Fr(is)31 b Fp(R)p Fr(-line)l(ar)f(for)g(e)l(ach)h Fp(~)f Fr(and)g Fn(Q)1623 431 y Fl(0)1690 461 y Fr(is)g(just)f(the)h (inclusion)g(map,)-118 561 y(and)g(the)g(fol)t(lowing)i(axioms)f(ar)l (e)f(ful\014l)t(le)l(d:)-118 660 y(\(a\))g(RIEFFEL'S)g(CONDITION:)g(F) -6 b(or)30 b(e)l(ach)60 b Ft(')24 b Fn(2)f Fq(A)1628 672 y Fl(0)1665 660 y Fr(,)30 b(the)g(map)g Ft(I)g Fn(3)24 b Fp(~)f Fn(!k)f(Q)2468 630 y Fo(~)2509 660 y Fv(\()p Ft(')p Fv(\))i Fn(k)2693 672 y Fo(~)2763 660 y Fn(2)g Fp(R)2902 672 y Fl(+)2986 660 y Fr(is)30 b(c)l(ontinuous.)-118 760 y(\(b\))g(V)n(ON)e(NEUMANN)h(CONDITION:)h(F)-6 b(or)30 b(e)l(ach)60 b Ft(';)14 b( )26 b Fn(2)e Fq(A)1934 772 y Fl(0)1971 760 y Fr(,)59 b Fn(k)23 b(Q)2188 730 y Fo(~)2229 760 y Fv(\()p Ft(')p Fv(\))c Ft(?)2408 730 y Fo(~)2467 760 y Fn(Q)2535 730 y Fo(~)2576 760 y Fv(\()p Ft( )s Fv(\))g Fn(\000)f(Q)2867 730 y Fo(~)2907 760 y Fv(\()p Ft(')h Fn(\016)f Ft( )s Fv(\))24 b Fn(k)3227 772 y Fo(~)3267 760 y Fn(!)f Fv(0)30 b Fr(when)g Fp(~)23 b Fn(!)g Fv(0)p Fr(.)-118 860 y(\(c\))30 b(DIRA)n(C'S)f(CONDITION:)g(F)-6 b(or)30 b(e)l(ach)61 b Ft(';)14 b( )26 b Fn(2)d Fq(A)1626 872 y Fl(0)1663 860 y Fr(,)60 b Fn(k)23 b(fQ)1923 830 y Fo(~)1963 860 y Fv(\()p Ft(')p Fv(\))p Ft(;)14 b Fn(Q)2186 830 y Fo(~)2227 860 y Fv(\()p Ft( )s Fv(\))p Fn(g)2390 872 y Fo(~)2450 860 y Fn(\000)k(Q)2601 830 y Fo(~)2655 860 y Fv(\()q Fn(f)p Ft(';)c( )s Fn(g)p Fv(\))22 b Fn(k)3016 872 y Fo(~)3057 860 y Fn(!)h Fv(0)29 b Fr(when)h Fp(~)23 b Fn(!)g Fv(0)p Fr(.)-118 959 y(\(e\))30 b(COMPLETENESS:)61 b Fn(Q)860 929 y Fo(~)901 959 y Fv(\()p Fq(A)993 971 y Fl(0)1030 959 y Fv(\))30 b Fr(is)g(dense)g(in)g Fq(C)1563 929 y Fo(~)1563 982 y(R)1638 959 y Fr(for)h(al)t(l)g Fp(~)23 b Fn(2)g Ft(I)7 b Fr(.)-118 1141 y Fv(The)18 b(w)n(ord)g(\\strict")f(w)n(as)h(coined)g(b)n(y)g(Rie\013el)h(in)g (order)e(to)i(distinguish)f(his)h(framew)n(ork)d(from)i(the)h (\(deformation\))g(quan)n(tization)-118 1240 y(de\014ned)28 b(in)g(terms)f(of)h(formal)e(series.)36 b(Usually)27 b(P)n(oisson)f(algebras)g(are)g(function)i(spaces:)-118 1422 y Fw(De\014nition)j(1.3)40 b Fr(We)c(c)l(al)t(l)i Fv(P)n(oisson)32 b(manifold)k Fr(a)h(smo)l(oth)f(manifold)i Ft(P)48 b Fr(so)37 b(that)f(on)g Ft(C)2763 1392 y Fs(1)2834 1422 y Fv(\()p Ft(P)12 b Fv(\))2963 1434 y Fo(R)3045 1422 y Fr(a)37 b(br)l(acket)f Fn(f\001)p Ft(;)14 b Fn(\001g)35 b Fr(is)i(given)-118 1522 y(such)30 b(that,)g(denoting)g(by)g Fn(\016)g Fr(the)g(p)l(ointwise)g(multiplic)l(ation,)i(the)e(triple)g Fv(\()q Ft(C)2288 1491 y Fs(1)2358 1522 y Fv(\()p Ft(P)12 b Fv(\))2487 1534 y Fo(R)2534 1522 y Ft(;)i Fn(\016)p Ft(;)g Fn(f\001)p Ft(;)g Fn(\001g)p Fv(\))28 b Fr(is)i(a)g(Poisson)h (algebr)l(a.)-118 1703 y Fv(When)k Ft(P)46 b Fv(is)35 b(not)f(compact,)i Ft(C)904 1673 y Fs(1)974 1703 y Fv(\()p Ft(P)12 b Fv(\))1103 1715 y Fo(R)1184 1703 y Fv(is)35 b(a)f(v)n(ery)f(large,)i(unnormed)f(space.)57 b(In)34 b(quan)n(tization)g(one)g(deals)g(with)h(suitable)-118 1803 y(families)27 b(of)h(smo)r(oth)f(b)r(ounded)h(observ)-5 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b(in)n(v)n(erse)-118 2273 y Ft(\014)-67 2238 y Fs(\000)p Fl(1)-71 2298 y Fm(\030)45 2273 y Fv(:)24 b(\004)f Fn(!)g Fv(\004)331 2243 y Fm(?)397 2273 y Fv(is)28 b(de\014ned)g(b)n(y)-3 2555 y(\()p Ft(\033)76 2567 y Fm(B)134 2555 y Fv(\))166 2570 y Fl(\()p Fm(q)r(;p)p Fl(\))308 2555 y Fv([\()p Ft(x;)14 b(k)s Fv(\))p Ft(;)g Fv(\()p Ft(y)s(;)g(l)r Fv(\)])24 b Fn(\021)f Ft(y)e Fn(\001)d Ft(k)j Fn(\000)d Ft(x)h Fn(\001)g Ft(l)h Fv(+)e Ft(x)h Fn(\001)p 1462 2489 68 4 v 18 w Ft(B)1529 2567 y Fm(q)1566 2555 y Ft(y)26 b Fv(=)1720 2463 y Fj(D)1771 2555 y Fv(\()p Ft(x;)14 b(k)s Fv(\))p Ft(;)g(\014)2053 2520 y Fs(\000)p Fl(1)2049 2584 y(\()p Fm(q)r(;p)p Fl(\))2192 2555 y Fv(\()p Ft(y)s(;)g(l)r Fv(\))2364 2463 y Fj(E)2428 2555 y Ft(;)70 b(\030)27 b Fv(=)22 b(\()p Ft(q)s(;)14 b(p)p Fv(\))p Ft(;)g(\021)27 b Fv(=)22 b(\()p Ft(x;)14 b(k)s Fv(\))p Ft(;)g(\020)31 b Fv(=)22 b(\()p Ft(y)s(;)14 b(l)r Fv(\))23 b Fn(2)g Fv(\004)p Ft(;)-118 2774 y Fv(where)p 123 2707 V 28 w Ft(B)191 2786 y Fm(q)253 2774 y Fv(:)i Ft(X)32 b Fn(!)25 b Ft(X)586 2744 y Fm(?)653 2774 y Fv(is)k(the)g(linear,)g(an) n(tisymmetric)f(op)r(erator)f(de\014ned)j(b)n(y)f Ft(B)2488 2786 y Fm(q)2524 2774 y Fv(\()p Ft(x;)14 b(y)s Fv(\))26 b(=)f Ft(x)20 b Fn(\001)p 2942 2707 V 20 w Ft(B)3009 2786 y Fm(q)3045 2774 y Ft(y)s Fv(,)30 b Fn(8)p Ft(x;)14 b(y)28 b Fn(2)e Ft(X)7 b Fv(.)40 b(It)30 b(follo)n(ws)-118 2874 y(easily)d(that)h Ft(\014)342 2844 y Fs(\000)p Fl(1)459 2874 y Fv(can)f(b)r(e)h(put)g(in)g(matrix-form)1044 3103 y Ft(\014)1095 3068 y Fs(\000)p Fl(1)1091 3128 y Fm(\030)1208 3103 y Fv(=)1295 2986 y Fj(\022)p 1398 2988 V 1398 3054 a Ft(B)1465 3066 y Fm(q)1586 3054 y Fn(\000)p Fv(1)1693 3066 y Fm(X)1752 3050 y Fh(\003)1398 3154 y Fv(1)1440 3166 y Fm(X)1667 3154 y Fv(0)1831 2986 y Fj(\023)1938 3103 y Fv(:)46 b Ft(X)25 b Fn(\002)18 b Ft(X)2260 3069 y Fm(?)2321 3103 y Fn(!)23 b Ft(X)2503 3069 y Fm(?)2559 3103 y Fn(\002)18 b Ft(X)r(;)-118 3333 y Fv(whic)n(h)27 b(leads)g(to)h(the)g(next)g(matrix-form)e(of)i Ft(\014)1367 3345 y Fm(\030)1403 3333 y Fv(:)1068 3570 y Ft(\014)1115 3582 y Fm(\030)1174 3570 y Fv(=)1262 3453 y Fj(\022)1446 3518 y Fv(0)164 b(1)1694 3530 y Fm(X)1365 3621 y Fn(\000)p Fv(1)1472 3633 y Fm(X)1531 3617 y Fh(\003)p 1652 3554 V 1652 3621 a Ft(B)1719 3633 y Fm(q)1798 3453 y Fj(\023)1910 3570 y Fv(:)46 b Ft(X)2055 3536 y Fm(?)2111 3570 y Fn(\002)18 b Ft(X)29 b Fn(!)23 b Ft(X)i Fn(\002)18 b Ft(X)2651 3536 y Fm(?)2688 3570 y Ft(:)-118 3838 y Fv(Th)n(us,)27 b(writting)h Ft(d)477 3850 y Fm(\030)513 3838 y Ft(h)23 b Fv(=)672 3746 y Fj(\020)721 3838 y Ft(d)764 3808 y Fm(X)764 3861 y(\030)828 3838 y Ft(h;)14 b(d)956 3808 y Fm(X)1015 3782 y Fi(?)956 3861 y Fm(\030)1054 3838 y Ft(h)1102 3746 y Fj(\021)1151 3838 y Fv(,)28 b(one)f(gets)867 4062 y Fn(f)p Ft(f)t(;)14 b(g)s Fn(g)1076 4074 y Fm(B)1132 4062 y Fv(\()p Ft(\030)t Fv(\))23 b(=)g Ft(d)1390 4027 y Fm(X)1390 4082 y(\030)1453 4062 y Ft(f)k Fn(\001)19 b Ft(d)1606 4027 y Fm(X)1665 4002 y Fi(?)1606 4082 y Fm(\030)1704 4062 y Ft(g)i Fn(\000)d Ft(d)1891 4027 y Fm(X)1891 4082 y(\030)1954 4062 y Ft(g)j Fn(\001)e Ft(d)2100 4027 y Fm(X)2159 4002 y Fi(?)2100 4082 y Fm(\030)2198 4062 y Ft(f)27 b Fv(+)18 b Ft(B)2412 4074 y Fm(q)2462 3994 y Fj(\000)2500 4062 y Ft(d)2543 4027 y Fm(X)2543 4082 y(\030)2607 4062 y Ft(f)t(;)c(d)2732 4027 y Fm(X)2732 4082 y(\030)2795 4062 y Ft(g)2838 3994 y Fj(\001)2889 4062 y Ft(:)-118 4266 y Fv(Using)36 b(co)r(ordinates,)g(one)g(has)g Ft(d)959 4236 y Fm(X)959 4290 y(\030)1022 4266 y Ft(h)h Fv(=)1208 4204 y Fj(P)1296 4224 y Fm(N)1296 4291 y(j)s Fl(=1)1429 4199 y Fj(\000)1467 4266 y Ft(@)1511 4278 y Fm(p)1545 4286 y Fi(j)1580 4266 y Ft(h)1628 4199 y Fj(\001)1680 4266 y Fv(\()p Ft(\030)t Fv(\))g Ft(dp)1906 4278 y Fm(j)1977 4266 y Fv(and)f Ft(d)2190 4236 y Fm(X)2249 4211 y Fi(?)2190 4290 y Fm(\030)2288 4266 y Ft(h)h Fv(=)2475 4204 y Fj(P)2562 4224 y Fm(N)2562 4291 y(j)s Fl(=1)2695 4199 y Fj(\000)2733 4266 y Ft(@)2777 4278 y Fm(q)2807 4286 y Fi(j)2843 4266 y Ft(h)2891 4199 y Fj(\001)2942 4266 y Fv(\()p Ft(\030)t 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b(shall)g(denote)g(b)n(y)f Fn(P)693 4891 y Fm(B)750 4879 y Fv(\(\004\))i(the)f(P)n(oisson)e(algebra)g Ft(C)1729 4849 y Fs(1)1800 4879 y Fv(\(\004\))1919 4891 y Fo(R)2002 4879 y Fv(endo)n(w)n(ed)h(with)i(the)f(p)r(oin)n(t)n(wise)f (m)n(ultiplication)i(and)e(the)-118 4979 y(P)n(oisson)27 b(brac)n(k)n(et)g Fn(f\001)p Ft(;)14 b Fn(\001g)644 4991 y Fm(B)729 4979 y Fv(giv)n(en)28 b(in)h(\(2\).)41 b(The)29 b(di\013eren)n(t)f(descriptions)g(of)h(the)g(quan)n(tum)g(observ)-5 b(able)28 b(algebras)e(in)j(the)h(next)-118 5078 y(Section)j(asks)f (also)g(for)g(a)h(partial)f(F)-7 b(ourier)32 b(transformed)g(v)n (ersion)f(of)i(this)h(P)n(oisson)d(algebra.)51 b(This)33 b(will)g(b)r(e)h(explained)e(in)-118 5178 y(Section)27 b(4)h(under)f(fa)n(v)n(orable)e(circumstances.)1869 5989 y(4)p eop end %%Page: 5 5 TeXDict begin 5 4 bop -118 154 a Fu(3)135 b(The)44 b(quan)l(tum)h (magnetic)h(observ)-7 b(ables;)46 b(the)f Fg(C)2574 111 y Fn(\003)2619 154 y Fu(-algebras)h(for)f Ff(~)33 b Fe(6)p Fd(=)h(0)-118 336 y 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y(p)r(oten)n(tial)c(is)g(used)f(to)h (de\014ne)g(a)g(represen)n(tation)e(of)i(some)f(explicitely)h(gauge)f (in)n(v)-5 b(arian)n(t)27 b(structure.)37 b(Only)27 b(this)i(one)e (will)h(b)r(e)-118 934 y(used)f(in)h(the)g(pro)r(cess)f(of)g(quan)n (tization.)-118 1033 y(Let)20 b(us)f(\014x)h(some)f(v)-5 b(alue)20 b Fp(~)j Fn(6)p Fv(=)f(0)e(for)f(the)h(Planc)n(k)e(constan)n (t.)34 b(W)-7 b(e)20 b(w)n(ould)f(lik)n(e)g(to)h(justify)h(the)f (construction)f(of)g(a)h(corresp)r(ondence)-118 1133 y Ft(f)32 b Fn(7!)23 b Fq(Op)171 1096 y Fo(~)171 1153 y Fm(A)225 1133 y Fv(\()p Ft(f)9 b Fv(\))24 b(b)r(et)n(w)n(een)g (\(suitable\))h(complex)e(functions)i(de\014ned)f(on)g(the)g(phase)g (space)f(\004)h(and)g(op)r(erators.)34 b(T)-7 b(o)24 b(the)g(function)-118 1232 y(\()p Ft(q)s(;)14 b(p)p Fv(\))29 b Fn(7!)g Ft(q)243 1244 y Fm(j)309 1232 y Fv(one)h(w)n(an)n(ts)g(to)h (assign)f(the)h(op)r(erator)f Ft(Q)1608 1244 y Fm(j)1673 1232 y Fv(of)h(m)n(ultiplication)g(with)h Ft(q)2537 1244 y Fm(j)2603 1232 y Fv(and)f(to)g(\()p Ft(q)s(;)14 b(p)p Fv(\))29 b Fn(7!)f Ft(p)3238 1244 y Fm(j)3304 1232 y Fv(w)n(e)j(asso)r(ciate)e(the)-118 1332 y(\014rst-order)34 b(di\013eren)n(tial)i(op)r(erator)e(\005)1108 1302 y Fo(~)1108 1355 y Fm(A;j)1250 1332 y Fv(:=)j Fp(~)p Ft(P)1473 1344 y Fm(j)1532 1332 y Fn(\000)24 b Ft(A)1683 1344 y Fm(j)1718 1332 y Fv(\()p Ft(Q)p Fv(\))37 b(=)g Fn(\000)p Ft(i)p Fp(~)p Ft(@)2170 1344 y Fm(j)2228 1332 y Fn(\000)24 b Ft(A)2379 1344 y Fm(j)2414 1332 y Fv(.)62 b(The)36 b(di\016cult)n(y)h(of)f(de\014ning)g(a)g(functional)-118 1451 y(calculus)27 b Ft(f)32 b Fn(7!)23 b Fq(Op)485 1415 y Fo(~)485 1472 y Fm(A)539 1451 y Fv(\()p Ft(f)9 b Fv(\))23 b Fn(\021)g Ft(f)9 b Fv(\()p Ft(Q;)14 b Fv(\005)1011 1421 y Fo(~)1011 1474 y Fm(A)1065 1451 y Fv(\))28 b(for)f(these)g(2)p Ft(N)36 b Fv(op)r(erators)26 b(comes)h(from)g(their)h(high)f(degree)g (of)g(non-comm)n(utativit)n(y:)432 1651 y Ft(i)p Fv([)p Ft(Q)550 1663 y Fm(j)584 1651 y Ft(;)14 b(Q)687 1663 y Fm(k)728 1651 y Fv(])23 b(=)f(0)p 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Fo(~)-12 2066 y Fm(A)42 2043 y Fv(\()p Ft(q)s Fv(\))24 b(:=)f Ft(e)320 2013 y Fm(iq)r Fs(\001)p Fl(\005)444 1988 y Fc(~)444 2030 y Fi(A)495 1951 y Fj(\021)545 2102 y Fm(q)r Fs(2)p Fm(X)685 2043 y Fv(,)28 b(giv)n(en)f(explicitely)g(in)h(the)g(Hilb)r(ert)g(space)f Fn(H)d Fv(:=)f Ft(L)2359 2013 y Fl(2)2396 2043 y Fv(\()p Ft(X)7 b Fv(\))27 b(b)n(y)1280 2243 y Ft(U)1346 2208 y Fo(~)1337 2263 y Fm(A)1391 2243 y Fv(\()p Ft(q)s Fv(\))d(=)e Ft(e)1645 2208 y Fs(\000)p Fl(\()p Fm(i=)p Fo(~)p Fl(\)\000)1884 2216 y Fi(A)1931 2208 y Fl(\([)p Fm(Q;Q)p Fl(+)p Fo(~)p Fm(q)r Fl(]\))2270 2243 y Ft(e)2309 2208 y Fm(iq)r Fs(\001)p Fo(~)p Fm(P)2476 2243 y Ft(;)1293 b Fv(\(3\))-118 2387 y(where)27 b(\000)174 2399 y Fm(A)228 2387 y Fv(\([)p Ft(q)323 2357 y Fs(0)347 2387 y Ft(;)14 b(q)424 2357 y Fs(0)465 2387 y Fv(+)k Fp(~)p Ft(q)s Fv(]\))24 b(:=)822 2321 y Fj(R)862 2417 y Fl([)p Fm(q)913 2400 y Fh(0)935 2417 y Fm(;q)987 2400 y Fh(0)1010 2417 y Fl(+)p Fo(~)p Fm(q)r Fl(])1167 2387 y Ft(A)k 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b(sev)n(ere)c(con)n(train)n(ts)h(on)-118 1150 y Ft(B)33 b Fv(one)c(hop)r(es)f(to)h(quan)n(tize)g(larger)e(classes)g (of)i(classical)f(sym)n(b)r(ols.)40 b(W)-7 b(e)29 b(shall)g(study)g(a)f (simple,)i(con)n(v)n(enien)n(t)e(situation;)h(the)-118 1250 y(reader)f(could)g(w)n(ork)g(out)h(other)g(cases)f(for)g(himself.) 42 b(W)-7 b(e)30 b(de\014ne)f Fn(A)2044 1220 y Fs(1)2140 1250 y Fv(:=)c Fn(f)p Ft(a)g Fn(2)h(A)20 b(\\)g Ft(C)2671 1220 y Fs(1)2741 1250 y Fv(\()p Ft(X)7 b Fv(\))26 b Fn(j)f Ft(@)3004 1220 y Fm(\013)3051 1250 y Ft(a)h Fn(2)g(A)p Ft(;)43 b Fn(8)p Ft(\013)26 b Fn(2)g Fp(N)3600 1220 y Fm(N)3663 1250 y Fn(g)p Fv(;)j(it)h(is)-118 1349 y(a)d(subspace)g(of)g Fn(A)19 b(\\)g Ft(B)t(C)683 1319 y Fs(1)753 1349 y Fv(\()p Ft(X)7 b Fv(\).)37 b(T)-7 b(ak)n(e)27 b Fn(A)1220 1361 y Fl(0)1280 1349 y Fv(=)c Fn(A)1434 1319 y Fs(1)1532 1349 y Fv(and)28 b Fn(S)h Fv(=)23 b Fn(S)6 b Fv(\()p Ft(X)h Fv(\),)28 b(the)g(Sc)n(h)n(w)n(artz)e(space)h(of)g(functions)h (on)f Ft(X)34 b Fv(whic)n(h)28 b(ha)n(v)n(e)-118 1449 y(rapidly)f(deca)n(ying)f(deriv)-5 b(ativ)n(es)27 b(of)g(an)n(y)g (order.)35 b(Then)28 b Fn(F)1706 1419 y Fs(\000)p Fl(1)1795 1449 y Fn(S)i Fv(=)23 b Fn(S)6 b Fv(\()p Ft(X)2127 1419 y Fm(?)2165 1449 y Fv(\))28 b(is)f(the)h(Sc)n(h)n(w)n(artz)e(space)h (de\014ned)h(on)f Ft(X)3503 1419 y Fm(?)3541 1449 y Fv(.)-118 1549 y(W)-7 b(e)28 b(also)e(consider)h Fn(S)6 b Fv(\()p Ft(X)679 1519 y Fm(?)717 1549 y Fv(;)14 b Fn(A)820 1519 y Fs(1)891 1549 y Fv(\),)28 b(the)g(space)f(of)g(functions)h Ft(X)1867 1519 y Fm(?)1928 1549 y Fn(3)23 b Ft(p)g Fn(7!)g Ft(f)9 b Fv(\()p Ft(p)p Fv(\))23 b Fn(2)g(A)2500 1519 y Fs(1)2599 1549 y Fv(suc)n(h)k(that)h(for)f(an)n(y)g Ft(l)r(;)14 b(m)22 b Fn(2)h Fp(N)784 1708 y Fn(k)p Ft(f)9 b Fn(k)918 1720 y Fm(l;m)1044 1708 y Fv(:=)23 b(max)o Fn(f)h Fv(sup)1351 1777 y Fm(p)p Fs(2)p Fm(X)1489 1761 y Fi(?)1523 1708 y Fn(k)p Ft(p)1607 1673 y Fm(\013)1654 1708 y Fv(\()p Ft(@)1735 1673 y Fm(\014)1780 1708 y Ft(f)9 b Fv(\)\()p Ft(p)p Fv(\))p Fn(k)2010 1720 y Fs(A)2113 1708 y Fn(j)46 b(j)p Ft(\013)p Fn(j)24 b(\024)f Ft(l)r(;)14 b Fn(j)p Ft(\014)t Fn(j)22 b(\024)h Ft(m)p Fn(g)f Ft(<)h Fn(1)p Ft(:)-118 1922 y Fv(W)-7 b(e)28 b(remark)e(that)1165 2022 y Fn(S)6 b Fv(\()p Ft(X)1329 1987 y Fm(?)1367 2022 y Fv(;)14 b Fn(A)1470 1987 y Fs(1)1541 2022 y Fv(\))23 b Fn(\032)g Ft(C)1749 1987 y Fs(1)1819 2022 y Fv(\(\004\))d Fn(\\)f Fp(F)2083 1987 y Fs(\000)p Fl(1)2172 2022 y Fn(f)p Ft(L)2271 1987 y Fl(1)2307 2022 y Fv(\()p Ft(X)7 b Fv(;)14 b Fn(A)p Fv(\))p Fn(g)p Ft(:)-118 2157 y Fv(Then)28 b(w)n(e)f(ha)n(v)n (e)f(the)i(follo)n(wing)f(eviden)n(t)-118 2316 y Fw(Prop)s(osition)j (4.1)41 b Fr(Supp)l(ose)31 b(that)g(the)f(c)l(omp)l(onents)h(of)h(the)f (magnetic)g(\014eld)g Ft(B)k Fr(b)l(elong)d(to)e Fn(A)2925 2285 y Fs(1)2996 2316 y Fr(.)42 b(Then)31 b Fn(S)6 b Fv(\()p Ft(X)3444 2285 y Fm(?)3482 2316 y Fv(;)14 b Fn(A)3585 2285 y Fs(1)3656 2316 y Fv(\))3688 2328 y Fo(R)3765 2316 y Fr(is)31 b(a)-118 2415 y(Poisson)g(sub)l(algebr)l(a)f(of)h Fn(P)742 2427 y Fm(B)799 2415 y Fv(\(\004\))f Fr(and)h(a)f(dense)g (subset)f(of)i(the)f(self-adjoint)h(p)l(art)f(of)h(the)f(ab)l(elian)h Ft(C)3077 2385 y Fs(\003)3115 2415 y Fr(-algebr)l(a)g Fn(A)19 b(\012)f Ft(C)3653 2427 y Fl(0)3690 2415 y Fv(\()p Ft(X)3798 2385 y Fm(?)3836 2415 y Fv(\))p Fr(.)-118 2574 y Fv(W)-7 b(e)28 b(can)f(no)n(w)g(state)-118 2733 y Fw(Theorem)k(4.2)41 b(\(Main)35 b(result\))d Fr(Assume)g(that)h(the)g(c)l(omp)l(onents)g (of)g(the)g(magnetic)h(\014eld)f Ft(B)k Fr(b)l(elong)c(to)g Fn(A)3427 2703 y Fs(1)3497 2733 y Fr(.)48 b(Then)33 b(the)-118 2832 y(family)e(of)g(inje)l(ctions)607 2765 y Fj(\000)646 2832 y Fn(S)6 b Fv(\()p Ft(X)810 2802 y Fm(?)848 2832 y Fv(;)14 b Fn(A)951 2802 y Fs(1)1021 2832 y Fv(\))1053 2844 y Fo(R)1123 2832 y Ft(,)-14 b Fn(!)23 b Fq(B)1311 2802 y Fo(~)1311 2855 y Fs(A)1369 2765 y Fj(\001)1407 2866 y Fo(~)p Fs(2)p Fl([0)p Fm(;)p Fl(1])1646 2832 y Fr(is)30 b(a)g(strict)g(deformation)h(quantization.)-118 3018 y Fv(As)i(seen)g(in)g(Section)g(3,)g(one)g(ma)n(y)f(sa)n(y)g(that) h Fq(B)1442 2988 y Fo(~)1442 3041 y Fs(A)1533 3018 y Fv(is)g(a)f Ft(C)1761 2988 y Fs(\003)1800 3018 y Fv(-algebra)e(of)j (\(magnetic\))g(pseudo)r(di\013eren)n(tial)g(sym)n(b)r(ols)f(and)h(its) -118 3126 y(represen)n(ted)c(v)n(ersions)g Fq(Op)754 3089 y Fo(~)754 3146 y Fm(A)822 3059 y Fj(\000)860 3126 y Fq(B)933 3096 y Fo(~)933 3149 y Fs(A)991 3059 y Fj(\001)1056 3126 y Fn(\032)e Ft(B)t Fv(\()p Ft(L)1304 3096 y Fl(2)1342 3126 y Fv(\()p Ft(X)7 b Fv(\)\))30 b(are)f Ft(C)1750 3096 y Fs(\003)1789 3126 y Fv(-algebras)f(of)i(magnetic)g(pseudo)r (di\013eren)n(tial)g(op)r(erators.)43 b(It)30 b(will)-118 3225 y(b)r(e)e(more)f(con)n(v)n(enien)n(t)f(to)i(w)n(ork)e(in)i(the)g (other)f(realization,)f(that)i(of)f(t)n(wisted)h(crossed)e(pro)r (ducts.)37 b(There)27 b(are)f(t)n(w)n(o)h(reasons:)-118 3325 y(1.)73 b(There)39 b(exist)h(results)f(of)h([17)o(])g(and)g([15)o (])g(on)g(con)n(tin)n(uous)e(\014elds)i(of)g(t)n(wisted)g(crossed)e (pro)r(ducts)i(whic)n(h)g(lead)f(almost)-118 3425 y(immediately)28 b(to)f(Rie\013el's)h(condition.)-118 3524 y(2.)34 b(In)22 b(the)g(t)n(wisted)f(crossed)f(pro)r(duct)i(formalism)e(one)h(disp)r (oses)g(of)g(the)h(simple)g(norm)f Fn(k)h(\001)h(k)2767 3536 y Fl(1)2804 3524 y Fv(,)g(whic)n(h)e(will)h(b)r(e)g(v)n(ery)e(con) n(v)n(enien)n(t)-118 3624 y(in)28 b(c)n(hec)n(king)e(the)i(axioms)f(of) g(v)n(on)g(Neumann)h(and)f(Dirac.)-118 3724 y(Th)n(us)22 b(w)n(e)f(state)h(no)n(w)f(a)h(v)-5 b(arian)n(t)21 b(of)h(Theorem)f (4.2;)i(these)f(t)n(w)n(o)f(results)g(are)g(equiv)-5 b(alen)n(t)22 b(b)n(y)g(the)g(isomorphisms)e(de\014ned)j(b)n(y)e(the) -118 3823 y(partial)27 b(F)-7 b(ourier)28 b(transformation.)37 b(W)-7 b(e)29 b(need)g(\014rst)f(to)g(rewrite)g(the)h(magnetic)e(P)n (oisson)g(structure.)39 b(On)28 b Fn(S)6 b Fv(\()p Ft(X)h Fv(;)14 b Fn(A)p Fv(\))29 b(\(ob)n(vious)-118 3923 y(de\014nition\))f (w)n(e)f(set)h(b)n(y)f(transp)r(ort)g(of)g(structure)338 4122 y Ft(')19 b Fn(\005)453 4088 y Fl(0)508 4122 y Ft( )26 b Fv(:=)d Fp(F)764 4055 y Fj(\002)798 4122 y Fv(\()p Fp(F)881 4088 y Fs(\000)p Fl(1)970 4122 y Ft(')p Fv(\)\()p Fp(F)1139 4088 y Fs(\000)p Fl(1)1229 4122 y Ft( )s Fv(\))1318 4055 y Fj(\003)1422 4122 y Fv(and)55 b Fn(f)p Ft(';)14 b( )s Fn(g)1843 4088 y Fm(B)1923 4122 y Fv(:=)22 b Fp(F)2098 4055 y Fj(\002)2133 4122 y Fn(f)p Fp(F)2226 4088 y Fs(\000)p Fl(1)2314 4122 y Ft(';)14 b Fp(F)2456 4088 y Fs(\000)p Fl(1)2545 4122 y Ft( )s Fn(g)2644 4134 y Fm(B)2701 4055 y Fj(\003)2749 4122 y Ft(;)97 b(';)14 b( )26 b Fn(2)e(S)6 b Fv(\()p Ft(X)h Fv(;)14 b Fn(A)p Fv(\))p Ft(:)-118 4257 y Fv(A)28 b(simple)g(direct)f(calculation)g(giv)n(es)1148 4478 y(\()p Ft(')19 b Fn(\005)1295 4444 y Fl(0)1351 4478 y Ft( )s Fv(\)\()p Ft(q)s Fv(;)14 b Ft(x)p Fv(\))24 b(=)1740 4365 y Fj(Z)1786 4554 y Fm(X)1863 4478 y Ft(dy)30 b(')p Fv(\()p Ft(q)s Fv(;)14 b Ft(y)s Fv(\))p Ft( )s Fv(\()p Ft(q)s Fv(;)g Ft(x)20 b Fn(\000)e Ft(y)s Fv(\);)1161 b(\(9\))-118 4665 y Fn(\005)-76 4635 y Fl(0)-10 4665 y Fv(is)28 b(p)r(oin)n(wise)h(m)n(ultiplication)f(in)h(the)h(\014rst)e (v)-5 b(ariable)28 b(and)h(con)n(v)n(olution)e(in)i(the)g(second.)40 b(Sligh)n(tly)28 b(more)g(e\013ort)h(is)g(needed)-118 4765 y(to)e(pro)n(v)n(e)f(that)234 5025 y Fn(f)p Ft(';)14 b( )s Fn(g)466 4990 y Fm(B)546 5025 y Fv(=)22 b Fn(\000)p Ft(i)771 4921 y Fm(N)740 4946 y Fj(X)743 5122 y Fm(j)s Fl(=1)874 4932 y Fj(h)913 5025 y Fv(\()p Ft(Q)1011 4981 y Fl(\(2\))1011 5048 y Fm(j)1100 5025 y Ft(')p Fv(\))d Fn(\005)1247 4990 y Fl(0)1302 5025 y Fv(\()p Ft(@)1383 4981 y Fl(\(1\))1378 5048 y Fm(j)1473 5025 y Ft( )s Fv(\))f Fn(\000)g Fv(\()p Ft(@)1744 4981 y Fl(\(1\))1739 5048 y Fm(j)1834 5025 y Ft(')p Fv(\))h Fn(\005)1981 4990 y Fl(0)2036 5025 y Fv(\()p Ft(Q)2134 4981 y Fl(\(2\))2134 5048 y Fm(j)2223 5025 y Ft( )s Fv(\))2312 4932 y Fj(i)2370 5025 y Fn(\000)2507 4921 y Fm(N)2477 4946 y Fj(X)2453 5124 y Fm(j;k)q Fl(=1)2634 5025 y Ft(B)2697 5037 y Fm(j)s(k)2769 5025 y Fv(\()p Fn(\001)p Fv(\)\()p Ft(Q)2954 4981 y Fl(\(2\))2954 5048 y Fm(j)3044 5025 y Ft(')p Fv(\))g Fn(\005)3191 4990 y Fl(0)3246 5025 y Fv(\()p Ft(Q)3344 4981 y Fl(\(2\))3344 5050 y Fm(k)3433 5025 y Ft( )s Fv(\))p Ft(;)205 b Fv(\(10\))-118 5283 y(where)122 5191 y Fj(\020)172 5283 y Ft(Q)238 5240 y Fl(\(2\))238 5306 y Fm(j)326 5283 y Ft(\032)369 5191 y Fj(\021)433 5283 y Fv(\()p Ft(q)s Fv(;)14 b Ft(x)p Fv(\))24 b(:=)e Ft(x)802 5295 y Fm(j)838 5283 y Ft(\032)p Fv(\()p Ft(q)s Fv(;)14 b Ft(x)p Fv(\))28 b(and)g(\()p Ft(@)1340 5240 y Fl(\(1\))1335 5306 y Fm(j)1429 5283 y Ft(\032)p Fv(\)\()p Ft(q)s Fv(;)14 b Ft(x)p Fv(\))24 b(=)1844 5251 y Fm(@)p 1814 5265 101 4 v 1814 5312 a(@)t(q)1883 5320 y Fi(j)1924 5283 y Ft(\032)p Fv(\()p Ft(q)s Fv(;)14 b Ft(x)p Fv(\).)-118 5413 y(Let)25 b(us)h(denote)f(b)n(y)g Ft(C)576 5383 y Fs(\003)614 5413 y Fv(\()p Ft(X)7 b Fv(\))26 b(the)f(group)g Ft(C)1219 5383 y Fs(\003)1257 5413 y Fv(-algebra)e(of)j Ft(X)7 b Fv(;)25 b(it)h(is)f(the)h(en)n(v)n (elopping)e Ft(C)2610 5383 y Fs(\003)2648 5413 y Fv(-algebra)g(of)h Ft(L)3114 5383 y Fl(1)3151 5413 y Fv(\()p Ft(X)7 b Fv(\),)25 b(the)h(con)n(v)n(olution)-118 5513 y(Banac)n(h)185 5483 y Fs(\003)223 5513 y Fv(-algebra)32 b(of)i Ft(X)7 b Fv(.)55 b(It)35 b(is)f(isomorphic)e(to)i Ft(C)1583 5525 y Fl(0)1621 5513 y Fv(\()p Ft(X)1729 5483 y Fm(?)1767 5513 y Fv(\))g(b)n(y)g(an)g (extension)f(of)h(the)h(F)-7 b(ourier)33 b(transformation;)i(th)n(us)f (the)-118 5641 y(sp)r(ectrum)25 b(of)h Ft(C)400 5610 y Fs(\003)438 5641 y Fv(\()p Ft(X)7 b Fv(\))25 b(is)h(homeomorphic)e (to)h Ft(X)1416 5610 y Fm(?)1453 5641 y Fv(.)37 b(Note)25 b(that)h(the)f(t)n(wisted)h(crossed)e(pro)r(duct)h Fq(C)2961 5610 y Fo(~)2961 5663 y Fs(A)3042 5641 y Fv(=)d Fn(A)14 b Fp(o)3274 5595 y Fm(!)3318 5570 y Fc(~)3316 5612 y Fi(B)3274 5671 y Fm(\022)3308 5654 y Fc(~)3383 5641 y Ft(X)32 b Fv(collapses)24 b(to)-118 5740 y Fn(A)19 b(\012)f Ft(C)115 5710 y Fs(\003)153 5740 y Fv(\()p Ft(X)7 b Fv(\))28 b(for)f Fp(~)c Fv(=)f(0.)1869 5989 y(8)p eop end %%Page: 9 9 TeXDict begin 9 8 bop -118 154 a Fw(Prop)s(osition)30 b(4.3)41 b Fr(Supp)l(ose)36 b(that)h(the)f(c)l(omp)l(onents)h(of)g(the) g(magnetic)g(\014eld)g Ft(B)j Fr(b)l(elong)d(to)g Fn(A)2988 124 y Fs(1)3058 154 y Fr(;)j(then)d(the)f(ve)l(ctor)h(sp)l(ac)l(e)-118 254 y Fn(S)6 b Fv(\()p Ft(X)h Fv(;)14 b Fn(A)149 223 y Fs(1)220 254 y Fv(\))252 266 y Fo(R)321 254 y Fv(=)23 b Fp(F)14 b Fv([)o Fn(S)6 b Fv(\()p Ft(X)660 223 y Fm(?)698 254 y Fv(;)14 b Fn(A)801 223 y Fs(1)872 254 y Fv(\)])28 b Fr(is)h(a)f(Poisson)i(algebr)l(a)f(for)g(the)f(c)l(omp)l(osition)i (laws)f(\(9\))f(and)h(\(10\).)39 b(It)27 b(is)i(also)g(dense)f(in)g (the)-118 353 y(self-adjoint)k(p)l(art)e(of)g(the)g(ab)l(elian)h Ft(C)1066 323 y Fs(\003)1105 353 y Fr(-algebr)l(a)f Fn(A)19 b(\012)f Ft(C)1648 323 y Fs(\003)1687 353 y Fv(\()p Ft(X)7 b Fv(\))p Fr(.)-118 519 y Fv(The)28 b(partial)e(F)-7 b(ourier)27 b(transformed)f(v)n(ersion)g(of)i(our)f(Main)g(Result)h (reads)-118 702 y Fw(Theorem)j(4.4)41 b Fr(Assume)26 b(that)h(the)g(c)l(omp)l(onents)g(of)h(the)f(magnetic)g(\014eld)h Ft(B)j Fr(b)l(elong)c(to)g Fn(A)2766 672 y Fs(1)2837 702 y Fr(;)h(then)f(the)g(family)i(of)f(inje)l(ctions)-118 734 y Fj(\000)-80 802 y Fn(S)6 b Fv(\()p Ft(X)h Fv(;)14 b Fn(A)187 771 y Fs(1)258 802 y Fv(\))290 814 y Fo(R)359 802 y Ft(,)-14 b Fn(!)23 b Fv(\()p Fq(C)557 771 y Fo(~)557 824 y Fs(A)615 802 y Fv(\))647 814 y Fo(R)693 734 y Fj(\001)731 835 y Fo(~)p Fs(2)p Fl([0)p Fm(;)p Fl(1])970 802 y Fr(is)30 b(a)h(strict)e(deformation)j(quantization.)-118 1011 y Fv(The)19 b(completeness)f(condition)h(is)g(ob)n(vious:)31 b Fn(S)6 b Fv(\()p Ft(X)h Fv(;)14 b Fn(A)1560 981 y Fs(1)1631 1011 y Fv(\))19 b(is)g(dense)g(in)2063 944 y Fj(\000)2101 1011 y Ft(L)2158 981 y Fl(1)2195 1011 y Fv(\()p Ft(X)7 b Fv(;)14 b Fn(A)p Fv(\))p Ft(;)g Fn(k)23 b(\001)g(k)2628 1023 y Fl(1)2664 944 y Fj(\001)2702 1011 y Fv(,)e Ft(L)2803 981 y Fl(1)2840 1011 y Fv(\()p Ft(X)7 b Fv(;)14 b Fn(A)p Fv(\))19 b(is)g(dense)g(in)3483 944 y Fj(\000)3521 1011 y Fq(C)3572 981 y Fo(~)3572 1034 y Fs(A)3630 1011 y Ft(;)14 b Fn(k)22 b(\001)h(k)3819 1023 y Fo(~)3860 944 y Fj(\001)-118 1111 y Fv(and)30 b(one)g(has)g Fn(k)e(\001)g(k)515 1123 y Fl(1)582 1111 y Fn(\024)i(k)e(\001)g(k)840 1123 y Fo(~)880 1111 y Fv(.)46 b(The)30 b(conditions)g(of)h(De\014nition)g(1.5)f(are)f (also)h(clearly)f(satis\014ed.)45 b(W)-7 b(e)31 b(still)g(ha)n(v)n(e)e (to)h(v)n(erify)-118 1210 y(the)e(conditions)f(\(a\),)h(\(b\))g(and)g (\(c\))g(of)f(De\014nition)i(1.2.)36 b(This)27 b(will)h(b)r(e)g(done)f (in)h(the)g(next)g(sections.)-118 1310 y Fw(Remark.)37 b Fv(It)29 b(w)n(ould)e(b)r(e)h(in)h(the)f(spirit)g(of)f(man)n(y)h(w)n (orks)e(in)i(strict)g(deformation)f(quan)n(tization)g(to)h(consider)f (only)g(the)i(case)-118 1410 y Fn(A)k Fv(=)g Ft(C)138 1422 y Fl(0)175 1410 y Fv(\()p Ft(X)7 b Fv(\).)55 b(Since)33 b(in)h(this)g(case)f Fq(C)1119 1380 y Fo(~)1119 1436 y Fm(C)1167 1444 y Fb(0)1198 1436 y Fl(\()p Fm(X)5 b Fl(\))1347 1410 y Fv(is)33 b(isomorphic)g(to)g Ft(K)2060 1342 y Fj(\002)2095 1410 y Ft(L)2152 1380 y Fl(2)2189 1410 y Fv(\()p Ft(X)7 b Fv(\))2329 1342 y Fj(\003)2363 1410 y Fv(,)35 b(the)f Ft(C)2635 1380 y Fs(\003)2674 1410 y Fv(-algebra)d(of)i(all)h(compact)f(op)r(erators)-118 1527 y(on)k Ft(L)64 1497 y Fl(2)101 1527 y Fv(\()p Ft(X)7 b Fv(\))38 b(\(cf.)68 b([12)o(],)41 b(Prop)r(osition)36 b(2.17)h(\(b\)\),)k(in)d(fact)g(one)g(w)n(orks)e(with)i(a)g(\014eld)g (of)g Ft(C)2835 1497 y Fs(\003)2873 1527 y Fv(-algebras)e(with)i(t)n(w) n(o)f(t)n(yp)r(es)h(of)-118 1626 y(\014b)r(ers:)48 b Ft(C)209 1638 y Fl(0)247 1626 y Fv(\(\004\))35 b(for)e Fp(~)f Fv(=)h(0)g(and)h Ft(K)1042 1559 y Fj(\002)1076 1626 y Ft(L)1133 1596 y Fl(2)1170 1626 y Fv(\()p Ft(X)7 b Fv(\))1310 1559 y Fj(\003)1378 1626 y Fv(for)33 b Fp(~)g Fn(6)p Fv(=)g(0.)54 b(W)-7 b(e)34 b(think)g(that)g(b)r(oth)g(the)g(t)n (wisted)f(crossed)f(pro)r(duct)i Fq(C)3673 1596 y Fo(~)3673 1649 y Fs(A)3764 1626 y Fv(and)-118 1726 y(the)28 b(pseudo)r (di\013eren)n(tial)f(formalism)g(are)g(useful)h(for)g(arbitrary)-7 b(,)25 b(admissible)j Fn(A)p Fv(.)38 b(In)28 b([13)o(])g(it)g(is)f(sho) n(wn)g(ho)n(w)h(to)f(calculate)g(the)-118 1826 y(essen)n(tial)33 b(sp)r(ectrum)g(and)h(ho)n(w)f(to)g(get)h(lo)r(calization)e(results)h (for)g(generalized)g(Sc)n(hr\177)-42 b(odinger)32 b(op)r(erators)f (with)j(anisotropic)-118 1925 y(p)r(oten)n(tials)c(and)h(magnetic)g (\014elds.)46 b(The)31 b(anisotrop)n(y)e(is)i(tak)n(en)f(in)n(to)h (accoun)n(t)f(b)n(y)g(the)i(ab)r(elian)e(algebra)f Fn(A)i Fv(and)g(exploiting)-118 2025 y(the)d(structure)f(of)g(its)h(sp)r (ectrum)g(is)g(the)f(k)n(ey)g(of)h(the)g(pro)r(ofs.)-118 2125 y Fw(Remark.)35 b Fv(Let)24 b(us)g(p)r(oin)n(t)f(out)h(that)g(if)g (the)g(sp)r(ectrum)g(of)g Fn(A)g Fv(is)f(compact)h(\(and)f(that)h(is)g (alw)n(a)n(ys)e(the)i(case)f(in)h(the)g(applications)-118 2224 y(to)36 b(quan)n(tum)f(Hamiltonians,)j(where)d(w)n(e)g(exp)r(ect)h Fn(A)h Fv(to)e(ha)n(v)n(e)g(a)g(unit)n(y\),)k(then)d(the)g(comp)r(onen) n(ts)f(of)h(the)g(magnetic)f(\014eld)-118 2324 y Ft(B)i Fv(b)r(eing)d(of)f(t)n(yp)r(e)h Fn(A)f Fv(eviden)n(tly)h(imply)f(that)h (they)f(are)g(b)r(ounded)h(and)f(uniformly)g(con)n(tin)n(uous.)53 b(Th)n(us,)35 b(in)e(this)h(case)e(the)-118 2423 y(requiremen)n(t)23 b(that)i(the)f(comp)r(onen)n(ts)g(of)g Ft(B)29 b Fv(are)23 b(of)h(class)g Fn(A)1777 2393 y Fs(1)1871 2423 y Fv(\(i.e.)37 b(they)24 b(are)f(of)i(class)e Ft(C)2712 2393 y Fs(1)2782 2423 y Fv(\()p Ft(X)7 b Fv(\))25 b(and)f(together)f(with)i(all)f(their) -118 2523 y(deriv)-5 b(ativ)n(es)29 b(admit)i(con)n(tin)n(uous)f (extensions)g(to)g(the)h(sp)r(ectrum)g(of)f Fn(A)p Fv(\))i(is)e(rather) g(optimal.)45 b(If)31 b(w)n(e)g(allo)n(w)e(the)i(sp)r(ectrum)g(of)-118 2623 y Fn(A)h Fv(to)g(b)r(e)g(noncompact,)g(then)g(w)n(e)g(can)f(allo)n (w)g(un)n(b)r(ounded)h(magnetic)f(\014elds)h(with)g(comp)r(onen)n(ts)f (of)h(class)f Fn(A)h Fv(but)h(w)n(e)e(ha)n(v)n(e)-118 2722 y(to)k(replace)e Fn(A)345 2692 y Fs(1)451 2722 y Fv(with)i Fn(A)713 2692 y Fs(1)713 2743 y Fm(c)819 2722 y Fv(the)g(subalgebra)e(of)i(elemen)n(ts)f(of)h Fn(A)2008 2692 y Fs(1)2114 2722 y Fv(that)g(ha)n(v)n(e)f(compact)g(supp)r(ort)g (\(with)i(resp)r(ect)f(to)f(the)-118 2822 y(sp)r(ectrum)28 b(of)f Fn(A)p Fv(\).)-118 3097 y Fu(5)135 b(Rie\013el's)47 b(condition)-118 3278 y Fv(T)-7 b(o)36 b(v)n(erify)g(the)h(con)n(tin)n (uit)n(y)f(condition)h(con)n(tained)f(in)h(Rie\013el's)g(axiom)e(w)n(e) i(need)f(a)h(result)f(sa)n(ying)f(roughly)h(that,)j(under)-118 3378 y(certain)d(conditions,)j(the)e(t)n(wisted)g(crossed)e(pro)r(duct) i(of)g(a)f(group)g(with)h(the)g(sectional)f(algebra)f(of)i(a)f Ft(C)3342 3348 y Fs(\003)3381 3378 y Fv(-bundle)h(is)f(the)-118 3478 y(sectional)27 b(algebra)g(of)h(a)g Ft(C)746 3448 y Fs(\003)784 3478 y Fv(-bundle)h(of)f(t)n(wisted)g(crossed)f(pro)r (ducts.)39 b(This)28 b(can)g(b)r(e)g(found)h(in)f([17)o(])h(and)f([15)o (];)h(tec)n(hniques)f(of)-118 3577 y([18)o(])j(and)h([3)o(])g(are)e (also)g(relev)-5 b(an)n(t)31 b(here.)47 b(F)-7 b(or)31 b(us)g(the)h(most)f(con)n(v)n(enien)n(t)f(reference)g(is)i([15)o(],)g (from)f(whic)n(h)g(w)n(e)g(quote)g(sligh)n(tly)-118 3677 y(reform)n(ulated)26 b(the)i(de\014nition)g(and)g(the)g(result)f(b)r (elo)n(w.)-118 3860 y Fw(De\014nition)k(5.1)40 b Fv(A)31 b(con)n(tin)n(uous)f Ft(C)1068 3829 y Fs(\003)1107 3860 y Fv(-bundle)i Fr(is)h(a)h(triple)f Fw(A)c Fv(=)1995 3792 y Fj(\000)2033 3860 y Ft(I)7 b(;)14 b Fn(fA)2221 3829 y Fo(~)2261 3860 y Fn(g)2303 3872 y Fo(~)p Fs(2)p Fm(I)2423 3860 y Ft(;)g Fv(\000)2512 3872 y Fl(0)2549 3860 y Fv(\()p Fw(A)p Fv(\))2685 3792 y Fj(\001)2723 3860 y Fr(,)34 b(wher)l(e)g Ft(I)39 b Fr(is)33 b(a)h(Hausdor\013,)g(lo) l(c)l(al)t(ly)-118 3959 y(c)l(omp)l(act)c(sp)l(ac)l(e,)h Fn(A)510 3929 y Fo(~)581 3959 y Fr(is)f(a)g Ft(C)807 3929 y Fs(\003)846 3959 y Fr(-algebr)l(a)h(with)f(norm)g Fn(k)22 b(\001)h(k)1711 3971 y Fo(~)1781 3959 y Fr(and)31 b Fv(\000)1995 3971 y Fl(0)2032 3959 y Fv(\()p Fw(A)p Fv(\))f Fr(a)g Ft(C)2335 3929 y Fs(\003)2374 3959 y Fr(-algebr)l(a)h (of)f(se)l(ctions)g(such)g(that:)-118 4059 y(\(i\))g(F)-6 b(or)30 b(any)g Fp(~)23 b Fn(2)g Ft(I)7 b Fr(,)60 b Fn(f)p Ft(F)12 b Fv(\()p Fp(~)p Fv(\))23 b Fn(j)g Ft(F)35 b Fn(2)24 b Fv(\000)1097 4071 y Fl(0)1134 4059 y Fv(\()p Fw(A)p Fv(\))p Fn(g)f Fv(=)g Fn(A)1489 4029 y Fo(~)1530 4059 y Fr(.)-118 4158 y(\(ii\))30 b(F)-6 b(or)30 b(any)h Ft(F)j Fn(2)24 b Fv(\000)563 4170 y Fl(0)600 4158 y Fv(\()p Fw(A)p Fv(\))p Fr(,)31 b(the)f(map)60 b Fp(~)23 b Fn(7!k)g Ft(F)35 b Fn(k)1488 4170 y Fo(~)1558 4158 y Fr(b)l(elongs)30 b(to)g Ft(C)2001 4170 y Fl(0)2038 4158 y Fv(\()p Ft(I)7 b Fv(\))p Fr(.)-118 4258 y(\(iii\))28 b Fv(\000)105 4270 y Fl(0)143 4258 y Fv(\()p Fw(A)p Fv(\))g Fr(is)f(a)h Ft(C)522 4270 y Fl(0)560 4258 y Fv(\()p Ft(I)7 b Fv(\))p Fr(-mo)l(dule:)38 b(if)28 b Ft(F)35 b Fn(2)23 b Fv(\000)1308 4270 y Fl(0)1345 4258 y Fv(\()p Fw(A)p Fv(\))28 b Fr(and)g Ft(\027)h Fn(2)23 b Ft(C)1875 4270 y Fl(0)1913 4258 y Fv(\()p Ft(I)7 b Fv(\))p Fr(,)28 b(then)g Ft(\027)5 b(F)39 b Fr(\(de\014ne)l(d)28 b(p)l(ointwise\))g(also)g(b)l(elongs)g(to)f Fv(\000)3698 4270 y Fl(0)3735 4258 y Fv(\()p Fw(A)p Fv(\))p Fr(.)-118 4441 y Fw(Theorem)k(5.2)41 b Fr(L)l(et)26 b Fw(A)h Fr(b)l(e)g(a)g(c)l(ontinuous)f Ft(C)1338 4411 y Fs(\003)1377 4441 y Fr(-bund)t(le)h(such)f(that)h Fv(\000)2073 4453 y Fl(0)2110 4441 y Fv(\()p Fw(A)p Fv(\))h Fr(is)f(sep)l(ar)l (able.)39 b(L)l(et)27 b Fv(\(\002)p Ft(;)14 b Fv(\012\))26 b Fr(b)l(e)h(a)g(twiste)l(d)g(action)h(of)-118 4540 y(an)j(amenable,)i (se)l(c)l(ond)f(c)l(ountable)f(lo)l(c)l(al)t(ly)i(c)l(omp)l(act)f(gr)l (oup)f Ft(X)38 b Fr(on)31 b Fv(\000)2109 4552 y Fl(0)2146 4540 y Fv(\()p Fw(A)p Fv(\))h Fr(by)g Ft(C)2483 4552 y Fl(0)2520 4540 y Fv(\()p Ft(I)7 b Fv(\))p Fr(-automorphisms.)45 b(Then)32 b(ther)l(e)f(exists)-118 4640 y(a)f(c)l(ontinuous)f Ft(C)437 4610 y Fs(\003)475 4640 y Fr(-bund)t(le)h Fw(C)23 b Fv(=)950 4573 y Fj(\000)988 4640 y Ft(I)7 b(;)14 b Fn(fC)1159 4610 y Fo(~)1199 4640 y Fn(g)1241 4652 y Fo(~)p Fs(2)p Fm(I)1360 4640 y Ft(;)g Fv(\000)1449 4652 y Fl(0)1486 4640 y Fv(\()p Fw(C)p Fv(\))1619 4573 y Fj(\001)1687 4640 y Fr(such)30 b(that:)-118 4761 y(\(i\))i(F)-6 b(or)33 b(any)f Fp(~)27 b Fn(2)h Ft(I)7 b Fr(,)65 b Fn(C)663 4731 y Fo(~)731 4761 y Fv(=)27 b Fn(A)889 4731 y Fo(~)950 4761 y Fp(o)1015 4731 y Fm(w)1065 4706 y Fc(~)1015 4790 y Fm(t)1040 4773 y Fi(h)1126 4761 y Ft(X)7 b Fr(,)32 b(wher)l(e)h Ft(t)1526 4731 y Fo(~)1526 4782 y Fm(x)1595 4761 y Fv(:)28 b Ft(X)33 b Fn(!)28 b Fv(Aut\()p Fn(A)2097 4731 y Fo(~)2139 4761 y Fv(\))p Fr(,)33 b Ft(t)2259 4731 y Fo(~)2259 4782 y Fm(x)2301 4761 y Fv([)p Ft(F)12 b Fv(\()p Fp(~)p Fv(\)])28 b(:=)f([\002)2752 4773 y Fm(x)2793 4761 y Fv(\()p Ft(F)12 b Fv(\)]\()p Fp(~)p Fv(\))p Fr(,)35 b Fn(8)p Ft(x)28 b Fn(2)f Ft(X)7 b Fr(,)33 b Fn(8)p Ft(F)39 b Fn(2)28 b Fv(\000)3724 4773 y Fl(0)3761 4761 y Fv(\()p Fw(A)p Fv(\))-118 4861 y Fr(and)i Ft(w)104 4831 y Fo(~)169 4861 y Fv(:)23 b Ft(X)i Fn(\002)18 b Ft(X)29 b 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b(=)f([\010\()p Ft(x)p Fv(\)]\()p Fp(~)p Fv(\))p Fr(,)31 b Fn(8)p Fp(~)23 b Fn(2)h Ft(I)7 b(;)14 b(x)23 b Fn(2)h Ft(X)7 b Fr(.)-118 5243 y Fw(Pro)s(of)41 b(of)h(Rie\013el's)c(condition.)61 b Fv(W)-7 b(e)36 b(are)f(placed)h (in)g(the)g(framew)n(ork)e(of)i(Section)g(3.)62 b(W)-7 b(e)36 b(start)f(b)n(y)h(constructing)f(a)-118 5342 y(t)n(wisted)g (action)g(on)f(a)h(large)e Ft(C)909 5312 y Fs(\003)948 5342 y Fv(-algebra,)i(consisting)f(of)h(functions)g(dep)r(ending)g(b)r (oth)h(on)f(the)g(v)-5 b(ariables)34 b Fp(~)h Fn(2)h Fv([0)p Ft(;)14 b Fv(1])34 b(and)-118 5442 y Ft(q)28 b Fn(2)d Ft(X)31 b Fn(\021)24 b Fp(R)276 5412 y Fm(N)339 5442 y Fv(.)39 b(The)29 b(same)f(strategy)f(has)h(b)r(een)h(used)g(in)g ([2)o(])g(for)f(the)h(rotation)e(algebras)g(\(whic)n(h)i(are)e(also)h (t)n(wisted)g(crossed)-118 5541 y(pro)r(ducts\))c(in)h(order)e(to)i (explore)e(the)i(regularit)n(y)e(of)h(the)h(sp)r(ectrum)g(of)f(certain) g(\014nite-di\013erence)h(op)r(erators,)e(the)i(parameter)-118 5641 y Fp(~)i 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5662 y Fv(\()p Ft(t;)14 b(s)p Fv(\))2331 5595 y Fj(\001)2393 5606 y Ft(@)5 b(\024)2490 5618 y Fm(j)p 2393 5643 132 4 v 2420 5719 a Ft(@)g(t)2545 5606 y(@)g(\024)2642 5618 y Fm(k)p 2545 5643 138 4 v 2570 5719 a Ft(@)g(s)1848 5989 y Fv(10)p eop end %%Page: 11 11 TeXDict begin 11 10 bop -118 154 a Fv(and)27 b(th)n(us)h(w)n(e)f(ha)n (v)n(e)-31 375 y Fn(\000)51 319 y Ft(i)p 44 356 45 4 v 44 432 a Fp(~)98 375 y Fv(\000)150 387 y Fm(B)207 375 y Fv(\()p Ft(\024)287 390 y Fm(q)r(;)p Fl(\()p Fm(x;y)r Fl(\))488 375 y Fv([\001)580 387 y Fl(2)618 375 y Fv(]\))c(=)g Fn(\000)p Ft(i)p Fp(~)937 296 y Fj(X)954 475 y Fm(j;k)1070 375 y Ft(y)1111 387 y Fm(j)1145 375 y Fv(\()p Ft(x)c Fn(\000)f Ft(y)s Fv(\))1402 387 y Fm(k)1457 262 y Fj(Z)1540 283 y Fl(1)1503 451 y(0)1591 375 y Ft(dt)1678 262 y Fj(Z)1761 283 y Fm(t)1724 451 y Fl(0)1804 375 y Ft(dsB)1949 387 y Fm(j)s(k)2035 258 y Fj(\024)2079 375 y Ft(q)j Fv(+)d(\()p Ft(s)h Fn(\000)2403 319 y Fv(1)p 2403 356 42 4 v 2403 432 a(2)2454 375 y(\))p Fp(~)p Ft(x)g 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a Fv(2)1898 1080 y Ft(y)s Fv(;)14 b Ft(x)k Fn(\000)g Ft(y)2171 963 y Fj(\023)2246 1080 y Ft(e)2285 1046 y Fs(\000)p Fm(i)p Fo(~)p Fl(\012)2444 1054 y Fi(B)2493 1046 y Fl(\()p Fm(q)r(;x;y)r Fl(;)p Fo(~)p Fl(\))2769 1080 y Fn(\000)g Ft(')p Fv(\()p Ft(q)s Fv(;)c Ft(y)s Fv(\))p Ft( )s Fv(\()p Ft(q)s Fv(;)g Ft(x)19 b Fn(\000)f Ft(y)s Fv(\))3482 960 y Fj(\014)3482 1009 y(\014)3482 1059 y(\014)3482 1109 y(\014)3524 1080 y Ft(:)203 b Fv(\(15\))-118 1285 y(It)28 b(is)f(easy)g(to)g(see)h(that)g(the)g(in)n(tegral)e(is)h (b)r(ounded)h(b)n(y)g(the)g(expression)468 1530 y(2)524 1413 y Fj(\022)590 1530 y Fv(sup)585 1600 y Fm(q)r Fs(2)p Fm(X)751 1530 y Fv(sup)744 1600 y Fm(y)r Fs(2)p Fm(X)906 1530 y Ft(<)23 b(y)j(>)1126 1496 y Fm(m)1211 1530 y Ft(')p Fv(\()p Ft(q)s Fv(;)14 b Ft(y)s Fv(\))1450 1413 y Fj(\023)h(\022)1593 1530 y Fv(sup)1587 1600 y Fm(q)r Fs(2)p Fm(X)1755 1530 y Fv(sup)1746 1600 y Fm(x)p Fs(2)p Fm(X)1911 1530 y Ft(<)22 b(x)i(>)2134 1496 y Fm(m)2220 1530 y Ft( )s Fv(\()p Ft(q)s Fv(;)14 b Ft(x)p Fv(\))2465 1413 y Fj(\023)h(\022)2602 1417 y(Z)2648 1605 y Fm(X)2725 1530 y Ft(dx)23 b(<)g(x)h(>)3062 1496 y Fs(\000)p Fm(m)3176 1413 y Fj(\023)3237 1430 y Fl(2)3288 1530 y Ft(;)-118 1773 y Fv(that)32 b(is)f(\014nite)h(and)f (not)h(dep)r(ending)g(of)f Fp(~)h Fv(for)e(an)n(y)h Ft(m)f(>)f(N)9 b Fv(.)48 b(On)31 b(the)h(other)f(hand)g(the)h(in)n(tegrand)f(in)g (\(15\))h(is)f(con)n(v)n(ergen)n(t)-118 1872 y(p)r(oin)n(t)n(wise)c(to) h(zero)e(when)i Fp(~)f Fv(go)r(es)g(to)g(0,)h(as)f(can)g(b)r(e)h(seen)f (after)g(writing)h(the)g(inequalit)n(y)413 2100 y(sup)408 2170 y Fm(q)r Fs(2)p Fm(X)558 1980 y Fj(\014)558 2030 y(\014)558 2079 y(\014)558 2129 y(\014)585 2100 y Ft(')653 1983 y Fj(\022)715 2100 y Ft(q)21 b Fn(\000)866 2044 y Fp(~)p 866 2081 V 868 2157 a Fv(2)921 2100 y(\()p Ft(x)e Fn(\000)f Ft(y)s Fv(\);)c Ft(y)1259 1983 y Fj(\023)1334 2100 y Ft( )1405 1983 y Fj(\022)1466 2100 y Ft(q)21 b Fv(+)1617 2044 y Fp(~)p 1617 2081 V 1619 2157 a Fv(2)1672 2100 y Ft(y)s Fv(;)14 b Ft(x)k 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y(\014)2342 2401 y Ft( )2412 2284 y Fj(\022)2474 2401 y Ft(q)21 b Fv(+)2625 2345 y Fp(~)p 2625 2382 V 2627 2458 a Fv(2)2680 2401 y Ft(y)s Fv(;)14 b Ft(x)k Fn(\000)g Ft(y)2953 2284 y Fj(\023)3014 2281 y(\014)3014 2331 y(\014)3014 2380 y(\014)3014 2430 y(\014)3042 2284 y(\023)3117 2401 y Fv(+)779 2669 y(+)858 2552 y Fj(\022)924 2669 y Fv(sup)919 2739 y Fm(q)r Fs(2)p Fm(X)1069 2669 y Fn(j)p Ft(')p Fv(\()p Ft(q)s Fv(;)c Ft(y)s Fv(\))p Fn(j)1354 2552 y Fj(\023)g(\022)1496 2669 y Fv(sup)1490 2739 y Fm(q)r Fs(2)p Fm(X)1640 2549 y Fj(\014)1640 2598 y(\014)1640 2648 y(\014)1640 2698 y(\014)1668 2669 y Ft( )1739 2552 y Fj(\022)1800 2669 y Ft(q)22 b Fv(+)1952 2613 y Fp(~)p 1952 2650 V 1954 2726 a Fv(2)2006 2669 y Ft(y)s Fv(;)14 b Ft(x)19 b Fn(\000)f Ft(y)2280 2552 y Fj(\023)2359 2669 y Fn(\000)g Ft( )s Fv(\()p Ft(q)s Fv(;)c Ft(x)19 b Fn(\000)f Ft(y)s Fv(\))2833 2549 y Fj(\014)2833 2598 y(\014)2833 2648 y(\014)2833 2698 y(\014)2861 2552 y(\023)2936 2669 y Fv(+)863 2937 y(+)942 2820 y Fj(\022)1008 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y(\015)1134 4214 y(\015)1206 4129 y Fv(1)p 1190 4166 74 4 v 1190 4242 a Ft(i)p Fp(~)1287 4118 y Fj(\000)1325 4185 y Ft(')19 b Fn(\005)1440 4150 y Fo(~)1499 4185 y Ft( )i Fn(\000)d Ft( )k Fn(\005)1775 4150 y Fo(~)1834 4185 y Ft(')1888 4118 y Fj(\001)1944 4185 y Fn(\000)c(f)p Ft(';)c( )s Fn(g)2259 4197 y Fm(B)2316 4064 y Fj(\015)2316 4114 y(\015)2316 4164 y(\015)2316 4214 y(\015)2362 4268 y Fl(1)2423 4185 y Fn(\000)-14 b(!)2422 4239 y Fo(~)p Fs(!)p Fl(0)2581 4185 y Fv(0)p Ft(:)1104 b Fv(\(16\))-118 4429 y(F)-7 b(or)22 b(that)h(w)n(e)g(shall)f(need)h(the)g(exact)f(form)h(of)g(the)g (\014rst)f(order)g(term)g(in)i Fp(~)e Fv(of)46 b Ft(')9 b Fn(\005)2419 4399 y Fo(~)2469 4429 y Ft( )j Fn(\000)d Ft( )j Fn(\005)2717 4399 y Fo(~)2765 4429 y Ft(')p Fv(.)36 b(W)-7 b(e)23 b(use)g(T)-7 b(a)n(ylor)21 b(dev)n(elopmen)n(ts)394 4689 y Ft(')462 4571 y Fj(\022)524 4689 y Ft(q)g Fn(\000)675 4632 y Fp(~)p 675 4669 45 4 v 677 4745 a Fv(2)730 4689 y(\()p Ft(x)e Fn(\000)f Ft(y)s Fv(\);)c Ft(y)1068 4571 y 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Fv(\)\()p Ft(q)s Fv(;)14 b Ft(x)p Fv(\))24 b(:=)1540 5636 y(1)p 1540 5673 42 4 v 1540 5749 a(2)1591 5692 y Ft(z)e Fn(\001)1693 5579 y Fj(Z)1776 5600 y Fl(1)1739 5768 y(0)1827 5692 y Ft(ds)1946 5600 y Fj(\020)1996 5692 y Fn(r)2065 5658 y Fl(\(1\))2154 5692 y Ft(\032)2197 5600 y Fj(\021)2261 5575 y(\022)2322 5692 y Ft(q)f Fn(\006)d Ft(s)2512 5636 y Fp(~)p 2512 5673 45 4 v 2514 5749 a Fv(2)2567 5692 y Ft(z)t Fv(;)c Ft(x)2694 5575 y Fj(\023)2768 5692 y Ft(:)959 b Fv(\(17\))1848 5989 y(11)p eop end %%Page: 12 12 TeXDict begin 12 11 bop -118 154 a Fv(Note)28 b(that)f(w)n(e)h(ha)n(v)n (e)1217 272 y(lim)1206 326 y Fo(~)p Fs(!)p Fl(0)1342 272 y Fv(\()p Fq(L)1429 238 y Fs(\006)1429 292 y Fm(z)1486 272 y Ft(')p Fv(\)\()p Ft(q)s Fv(;)14 b Ft(x)p Fv(\))24 b(=)1882 216 y(1)p 1882 253 42 4 v 1882 329 a(2)1934 272 y Ft(z)d Fn(\001)2036 180 y Fj(\020)2085 272 y Fn(r)2154 238 y Fl(\(1\))2244 272 y Ft(')2298 180 y Fj(\021)2361 272 y Fv(\()p Ft(q)s Fv(;)14 b Ft(x)p Fv(\))p Ft(:)1178 b Fv(\(18\))-118 450 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3624 y Ft(dy)j(y)1814 3636 y Fm(j)1849 3624 y Ft(')p Fv(\()p Ft(q)s Fv(;)14 b Ft(y)s Fv(\))24 b(\()p Ft(x)19 b Fn(\000)f Ft(y)s Fv(\))2369 3636 y Fm(k)2410 3624 y Ft( )s Fv(\()p Ft(q)s Fv(;)c Ft(x)19 b Fn(\000)f Ft(y)s Fv(\)+)414 3915 y(+)p Fp(~)538 3802 y Fj(Z)583 3990 y Fm(X)660 3915 y Ft(dy)784 3823 y Fj(h)823 3915 y Ft(y)s(')p Fv(\()p Ft(q)s Fv(;)c Ft(y)s Fv(\))19 b Fn(\001)f Fv(\()p Fn(r)1267 3880 y Fl(\(1\))1357 3915 y Ft( )s Fv(\)\()p Ft(q)s Fv(;)c Ft(x)19 b Fn(\000)f Ft(y)s Fv(\))h Fn(\000)f Fv(\()p Fn(r)1983 3880 y Fl(\(1\))2072 3915 y Ft(')p Fv(\)\()p Ft(q)s Fv(;)c Ft(y)s Fv(\))20 b Fn(\001)e Fv(\()p Ft(x)h Fn(\000)f Ft(y)s Fv(\))p Ft( )s Fv(\()p Ft(q)s Fv(;)c Ft(x)19 b Fn(\000)f Ft(y)s Fv(\))3052 3823 y Fj(i)3110 3915 y Fv(+)g Ft(o)p Fv(\()p Fp(~)p Fv(\))p Ft(;)-118 4118 y Fv(The)28 b(result)f(is)g(no)n(w)g(straigh)n (tforw)n(ard)e(b)n(y)i(the)h(explicit)g(form)f(of)h(the)g(brac)n(k)n (et)e Fn(f\001)p Ft(;)14 b Fn(\001g)2570 4088 y Fm(B)2654 4118 y Fv(and)27 b(of)h(the)g(comp)r(osition)f(la)n(w)g Fn(\005)3709 4088 y Fl(0)3745 4118 y Fv(.)-118 4393 y Fu(References)-35 4575 y Fv([1])41 b(F.)27 b(Ba)n(y)n(en,)f(M.)h (Flato,)g(C.)g(F)-7 b(ronsdal,)26 b(A.)h(Lic)n(hnero)n(wicz)e(and)i(D.) g(Sternheimer:)36 b Fr(Deformation)30 b(The)l(ory)h(and)f(Quanti-)94 4674 y(zation)p Fv(,)e(I,)g(I)r(I,)g(Ann.)g(Ph)n(ys.)f Fw(110)p Fv(,)g(61-11,)e(11-151,)g(\(1978\).)-35 4840 y([2])41 b(J.)c(Bellissard:)54 b Fr(Lipshitz)39 b(Continuity)g(of)g (Gap)g(Boundaries)h(for)f(Hofstadter-like)h(Sp)l(e)l(ctr)l(a)p Fv(,)f(Comm.)e(Math.)g(Ph)n(ys.)94 4940 y Fw(160)p Fv(,)27 b(599{613,)d(\(1994\).)-35 5106 y([3])41 b(E.)27 b(Blanc)n(hard:)36 b Fr(D)n(\023)-40 b(eformations)31 b(de)f Ft(C)1327 5076 y Fs(\003)1365 5106 y Fr(-alg)n(\022)-40 b(ebr)l(es)31 b(de)f(Hopf)p Fv(,)f(Bull.)f(So)r(c.)f(Math.)h(F)-7 b(rance)27 b Fw(124)p Fv(,)g(141-215,)d(\(1996\).)-35 5272 y([4])41 b(R.)f(Busb)n(y)e(and)i(H.)f(Smith:)62 b Fr(R)l(epr)l(esentations)40 b(of)h(Twiste)l(d)h(Gr)l(oup)f(A)n(lgebr)l(as)p Fv(,)i(T)-7 b(rans.)38 b(Amer.)i(Math.)f(So)r(c.)g Fw(149)p Fv(,)94 5372 y(503-537,)25 b(\(1970\).)-35 5538 y([5])41 b(M.)36 b(V.)g(Karasev)e(and)h(T.)h(A.)g(Osb)r(orn:)53 b Fr(Symple)l(ctic)38 b(A)n(r)l(e)l(as,)h(Quantization,)h(and)d(Dynamics)h(in)g(Ele)l(ctr)l (omagnetic)94 5637 y(Fields)p Fv(,)29 b(J.)f(Math.)f(Ph)n(ys.)g Fw(43)p Fv(,)g(756-788,)e(\(2002\).)1848 5989 y(12)p eop end %%Page: 13 13 TeXDict begin 13 12 bop -35 154 a Fv([6])41 b(M.)30 b(V.)h(Karasev)d (and)i(T.)g(A.)g(Osb)r(orn:)41 b Fr(Quantum)31 b(Magnetic)i(A)n(lgebr)l (a)f(and)h(Magnetic)g(Curvatur)l(e)p Fv(,)e(J.)f(Ph)n(ys.)f(A)h Fw(37)p Fv(,)94 254 y(2345{2363,)23 b(\(2004\).)-35 420 y([7])41 b(N.)23 b(P)-7 b(.)21 b(Landsman:)34 b Fr(Mathematic)l(al)26 b(T)-6 b(opics)27 b(Betwe)l(en)d(Classic)l(al)j(and)e(Quantum)e(Me)l (chanics)p Fv(,)i(Springer-V)-7 b(erlag,)21 b(New-)94 519 y(Y)-7 b(ork,)27 b(1998.)-35 685 y([8])41 b(N.)k(P)-7 b(.)45 b(Landsman)f(and)h(B.)g(Ramazan:)70 b Fr(Quantization)46 b(of)h(Poisson)f(A)n(lgebr)l(as)h(Asso)l(ciate)l(d)f(to)g(Lie)g(A)n (lgebr)l(oids)p Fv(,)94 785 y(Group)r(oids)25 b(in)g(analysis,)g (geometry)-7 b(,)25 b(and)g(ph)n(ysics)g(\(Boulder,)g(CO,)g(1999\),)g (159-192,)d(Con)n(temp.)k(Math.)f Fw(282)p Fv(,)h(Amer.)94 885 y(Math.)i(So)r(c.,)f(Pro)n(vidence,)f(RI,)i(2001.)-35 1051 y([9])41 b(J.)31 b(M.)h(Luttinger:)44 b Fr(The)35 b(E\013e)l(ct)e(of)h(the)f(Magnetic)i(Field)g(on)e(Ele)l(ctr)l(ons)g (in)h(a)f(Perio)l(dic)j(Potential)p Fv(,)d(Ph)n(ys.)e(Rev.)g Fw(84)p Fv(,)94 1150 y(814-817,)25 b(\(1951\).)-77 1316 y([10])41 b(M.)35 b(M\025)-42 b(an)n(toiu)33 b(and)i(R.)f(Purice:)50 b Fr(The)37 b(A)n(lgebr)l(a)f(of)h(Observables)g(in)f(a)g(Magnetic)i (Field)p Fv(,)f(Mathematical)d(Results)g(in)94 1416 y(Quan)n(tum)25 b(Mec)n(hanics)g(\(T)-7 b(axco,)25 b(2001\),)f(Con)n(temp)r(orary)f (Mathematics)i Fw(307)p Fv(,)h(Amer.)f(Math.)h(So)r(c.,)f(Pro)n (vidence,)f(RI,)94 1516 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Fv(,)h(595-612,)94 3176 y(\(1990\).)-77 3342 y([18])41 b(M.)23 b(Rie\013el:)35 b Fr(Continuous)24 b(Fields)j(of)f Ft(C)1340 3312 y Fs(\003)1378 3342 y Fr(-A)n(lgebr)l(as)g(Coming)g(fr)l(om)f(Gr)l(oup)h(Co)l(cycles)g (and)g(A)l(ctions)p Fv(,)d(Math.)g(Ann.)g Fw(283)p Fv(,)94 3442 y(631-643,)i(\(1989\).)-77 3608 y([19])41 b(M.)28 b(Rie\013el:)37 b Fr(Deformation)31 b(Quantization)f(for)g(A)l(ctions)g (of)h Fp(R)2087 3578 y Fm(d)2125 3608 y Fv(,)d(Memoirs)f(of)g(the)h (AMS,)g Fw(106)p Fv(,)f(\(1993\).)-77 3774 y([20])41 b(M.)g(Rie\013el:)64 b Fr(Quantization)42 b(and)g Ft(C)1299 3744 y Fs(\003)1337 3774 y Fr(-A)n(lgebr)l(as)p Fv(,)j Ft(C)1802 3744 y Fs(\003)1840 3774 y Fv(-algebras:)61 b(1943{1993)37 b(\(San)j(An)n(tonio,)k(TX,)d(1993\),)i(66-97,)94 3873 y(Con)n(temp.)28 b(Math.)f Fw(167)p Fv(,)g(Amer.)h(Math.)g(So)r (c.,)f(Pro)n(vidence,)f(RI,)i(1994.)-77 4039 y([21])41 b(I.)32 b(V)-7 b(aisman:)46 b Fr(L)l(e)l(ctur)l(es)33 b(on)h(the)g(Ge)l(ometry)g(of)h(Poisson)g(Manifolds)p Fv(,)h(Progress)29 b(in)j(Mathematics)g Fw(118)p Fv(,)h(Birkh\177)-42 b(auser)94 4139 y(V)-7 b(erlag,)27 b(Basel,)f(1994.)1848 5989 y(13)p eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF ---------------0411220746714--