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Deformations of gerbes on smooth manifolds 385corresponding Čech cohomology is trivial. Therefore, there exists 2 xC 1 .UI J 0 /such that c D L @. Then, 2 Isom 0 .A 01 ˝ J N1 U; J.A 01 //.Suppose that ; 0 2 Isom 0 .A 01 ˝ J N1 U; J.A 01 //. By Corollary 7.4 D 0for some uniquely defined 2 .N 2 UI J 0 /. It is easy to see that 2 xZ 1 .UI J 0 /.We assume from now on that we have chosen 2 Isom 0 .A 01 ˝ J N1 U; J.A 01 //,r2C .A 01 /. Such a choice defines p ij 2 Isom 0.A ij ˝ J Np U; J.A ij // for every pand 0 i;j p by p ij D .prp ij / . This collection of p ijinduces for every p algebraisomorphism p W Mat.A/ p ˝ J Np U ! Mat.J.A// p . The following compatibilityholds for these isomorphisms. Let f W Œp ! Œq be a morphism in . Then thefollowing diagram commutes:f .Mat.A/ p ˝ J Np U/f f ] Mat.A ˝ J Nq U/ q(7.4)f . p /f ] . p /f Mat.J.A// p f f ] Mat.J.A// q .Similarly define the connections r p ijD .pr p ij / r. For p D 0;1;::: set r p D˚pi;jD0 r ij ; the connections r p on Mat.A/ p satisfyf r p D .Ad f /.f ] r q /: (7.5)Note that F.;r/ 2 .N 1 UI 1 N 1 U ˝ J N 1 U/ is a cocycle of degree 1 in LC .UI 1 X ˝J X /. Vanishing of the corresponding Čech cohomology implies that there exists F 0 2.N 0 UI 1 N 0 U ˝ J N 0 U/ such that.d 1 / F 0 .d 0 / F 0 D F.;r/: (7.6)For p D 0;1;:::, 0 i p, let F piiD .pr p i / F 0 ; put F pij D 0 for i ¤ j .Let F p 2 .N p UI 1 N p U ˝ Mat.A/p / ˝ J Np U denote the diagonal matrix withcomponents F pij.Forf W Œp ! Œq we havef F p D f ] F q : (7.7)Then, we obtain the following equality of connections on Mat.A/ p ˝ J Np U:. p / 1 ır can ı p Dr p ˝ Id C Id ˝r can C ad F p : (7.8)The matrices F p also have the following property. Let r can F p be the diagonalmatrix with the entries .r can F p / ii Dr can F pii . Denote by rcan F p the image of r can F punder the natural map .N p UI 2 N p U ˝ Mat.A/p ˝ J Np U/ ! .N p UI 2 N p U ˝Mat.A/ p ˝ .J Np U=O Np U//. Recall the canonical map p W N p U ! X. Then, wehave the following:
386 P. Bressler, A. Gorokhovsky, R. Nest, and B. TsyganLemma 7.13. There exists a unique ! 2 .XI . 2 X ˝ J X=O X / cl / such thatr can F p D p ! ˝ Id p;where Id p denotes the .p C 1/ .p C 1/ identity matrix.Proof. Using the definition of F 0 and formula (7.2) we obtain: .d 1 / r can F 0.d 0 / r can F 0 Dr can F.;r/ 2 2 .N 1 U/. Hence .d 1 / r can F 0 .d 0 / r can F 0 D 0,and there exists a unique ! 2 .XI X 2 ˝ .J X=O X // such that 0 ! D rcan F 0 .Since .r can / 2 D 0 it follows that r can ! D 0. For any p we have: .r can F p / ii Dpr i rcan F 0 D p !, and the assertion of the lemma follows.Lemma 7.14. The class of ! in H 2 ..XI X ˝ J X=O X /; r can / does not depend onthe choices made in its construction.Proof. The construction of ! is dependent on the choice of F 0 , 2 Isom 0 .A 01 ˝J N1 UJ.A 01 //, and r 2 C .A 01 / satisfying the equation (7.6). Assume that wemake different choices: 0 D .@/ L , r 0 D .@˛/ L rand .F 0 / 0 satisfying @.F L 0 / 0 DF. 0 ; r 0 /. Here, 2 xC 0 .UI J 0 / and ˛ 2 xC 0 .UI 1 /. We have: F. 0 ; r 0 / DF.;r/ @˛ L C @r L can . It follows that @..F L 0 / 0 F 0 r can C ˛/ D 0. Therefore.F 0 / 0 F 0 r can C ˛ D 0ˇ for some ˇ 2 .XI 1 X ˝ J X/. Hence if ! 0 isconstructed using 0 , r 0 , .F 0 / 0 then ! 0 ! Dr can Nˇ where Nˇ is the image of ˇ underthe natural projection .XI X 1 ˝ J X/ ! .XI X 1 ˝ .J X=O X //.Let W V ! U be a refinement of the cover U, and .V; A / the correspondingdescent datum. Choice of , r, F 0 on U induces the corresponding choice .N/ ,.N/ r, .N/ .F 0 / on V. Let ! denotes the form constructed as in Lemma 7.13using .N/ , .N/ r, .N/ F 0 . Then,! D .N/ !:The following result now follows easily and we leave the details to the reader.Proposition 7.15. The class of ! in H 2 ..XI X ˝ J X=O X /; r can / coincides withthe image ŒS of the class of the gerbe.7.5 Construction of the quasiisomorphism. For W Œn ! letH WD .N .n/ UI N .n/ U ˝ .0n/ C .Mat.A/ .0/ ˝ J N.0/ U/ loc Œ1/;considered as a graded Lie algebra. For W Œm ! Œn, D ı there is a morphismof graded Lie algebras W H ! H .Forn D 0;1;:::let H n WD Q H . TheŒn! assignment 3 Œn 7! H n , 7! defines a cosimplicial graded Lie algebra H.For each W Œn ! the map WD Id ˝ .0n/ . .0/ /W H ! G DR .J.A//
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EMS Series of Congress ReportsEMS S
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Editors:Guillermo CortiñasDepartam
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ContentsPreface....................
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IntroductionSince its inception 50
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Introductionxiwith D. Blecher, he c
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Program list of speakers and topics
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Categorical aspects of bivariant K-
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42 A. Bartels, S. Echterhoff, and W
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72 H. Emerson and R. MeyerIn this s
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74 H. Emerson and R. MeyerK-theoret
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76 H. Emerson and R. MeyerExample 4
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78 H. Emerson and R. Meyerand exact
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80 H. Emerson and R. Meyercondition
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82 H. Emerson and R. MeyerAs a resu
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84 H. Emerson and R. MeyerLet h and
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86 H. Emerson and R. MeyerCorollary
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88 H. Emerson and R. Meyer5 Dual-Di
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On K 1 of a Waldhausen categoryFern
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On K 1 of a Waldhausen category 93t
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On K 1 of a Waldhausen category 95(
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On K 1 of a Waldhausen category 992
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118 M. Karoubi[35], M. F. Atiyah an
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120 M. Karoubihere by GrK n .A/, ar
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122 M. Karoubi(resp. 1) in A. In th
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124 M. Karoubithe class ˛ of A in
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126 M. Karoubiested in the complex
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128 M. KaroubiThe same method may b
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130 M. KaroubiBy the usual excision
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132 M. KaroubiLet A be a graded twi
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134 M. Karoubithe same ideas as in
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136 M. KaroubiTheorem 6.2. 25 The (
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138 M. KaroubiIn order to show this
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140 M. KaroubiThe following theorem
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142 M. KaroubiWe would like to poin
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144 M. KaroubiZ=n identified with t
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146 M. Karoubiwhere n is the ring
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148 M. Karoubi[6] M. F. Atiyah, R.
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Equivariant cyclic homology for qua
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174 C. Voigtand using ˇ.x; y/ D .S
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176 C. Voigtalgebra A. The space Cn
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178 C. VoigtKK-theory [16]. Hence,
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A Schwartz type algebra for the tan
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202 J. CuntzWe denote by N the set
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204 J. CuntzConsider the action of
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206 J. Cuntzprime numbers p and q,
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208 J. Cuntz6 Representations as cr
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210 J. CuntzProof. The spectral pro
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212 J. CuntzThe operator s 1 plays
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214 J. CuntzProposition 7.4. The K-
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On a class of Hilbert C*-manifoldsW
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On a class of Hilbert C*-manifolds
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228 U. Bunke, T. Schick, M. Spitzwe
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Page 349 and 350: 334 U. Bunke, T. Schick, M. Spitzwe
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List of contributorsArthur Bartels,
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List of participantsNatalia Abad Sa
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