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C -algebras associated with the ax C b-semigroup over N 207shows that 0 .e n / D 1=n and 0 .u/ D 0.The relation.s n s n/ D n.s n s n /and the 1-KMS-condition show that t .s n / D n it s n .5 The K-groups of Q NThe K-groups of Q N can be computed using the fact that Q N is a crossed product of theBunce–Deddens algebra F by the semigroup N . The K-groups of a Bunce–Deddensalgebra are well known and easy to determine using its representation as an inductivelimit of algebras of type M n .C.S 1 //. Specifically, for F ,wehaveK 0 .F / D Q; K 1 .F / D Z:We consider Q N as an inductive limit of the subalgebras B n D C .F ;s p1 ;:::;s pn /where 2 D p 1
208 J. Cuntz6 Representations as crossed productsFor each n, define the endomorphism ' n of Q N by ' n .x/ D s n xs n . Since ' n' m D ' nmthis defines an inductive system.Definition 6.1. We define xQ N as the inductive limit of the inductive system .Q N ;' n /.By construction we have a family n of natural inclusions of Q N into the inductivelimit xQ N satisfying the relations nm ' n D m . We denote by 1 k the element k .1/ ofxQ N . We have that 1 k D kl .e l / 1 kl . The union of the subalgebras 1 kxQ N 1 k is densein xQ N and 1 k 1 kl for all k; l. In order to define a multiplier a of xQ N it thereforesuffices to define a1 k and 1 k a for all k.We can extend the isometries s n naturally to unitaries Ns n in the multiplier algebraM. xQ N / of xQ N by requiringNs n 1 k D k .s n /; 1 k Ns n D kn .s n /:Note that this is well defined because, using the identities s n e l D e nl s n and ' l .s n / Ds n e l ,wehave.1 k Ns n /1 l D kn .s n / l .1/ D knl .s n e l e kn / D knl .e nl s n e kn / D k .1/ l .s n / D 1 k .Ns n 1 l /:Proposition 6.2. The elements Ns n , n 2 N, define unitaries in M. xQ N / such that Ns n Ns m DNs nm and such that Ns n Ns m DNs m Ns n.Defining Ns a DNs n Ns m for a D n=m 2 Q C we define unitaries in M. xQ N / such thatNs a Ns b DNs ab for a; b 2 Q C .Proof. In order to show that Ns n is unitary it suffices to show that 1 k Ns n Ns n1 k Ns n Ns n D 1 k for all k, n.D 1 k andWe can also extend the generating unitary u in Q N to a unitary in the multiplieralgebra M. xQ N /. We define the unitary Nu in M. xQ N / by the identityNu 1 k D k .u k /:We can also define fractional powers of Nu by settingNu 1=n 1 kn D kn .u k /:Proposition 6.3. For all a 2 Q Cand b 2 Q we have the identityProof. Check that Ns 1=n Nu DNu 1=n Ns 1=nNs a Nu b DNu ab Ns a :Following Bost–Connes we denote by P C Qthe ax C b-group P C 1 bQ D j a 2 Q 0 aC ;b2 Q :
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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 97s
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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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Page 172 and 173: Equivariant cyclic homology for qua
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Page 189 and 190: 174 C. Voigtand using ˇ.x; y/ D .S
Page 191 and 192: 176 C. Voigtalgebra A. The space Cn
Page 193 and 194: 178 C. VoigtKK-theory [16]. Hence,
Page 196 and 197: A Schwartz type algebra for the tan
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Page 217 and 218: 202 J. CuntzWe denote by N the set
Page 219 and 220: 204 J. CuntzConsider the action of
Page 221: 206 J. Cuntzprime numbers p and q,
Page 225 and 226: 210 J. CuntzProof. The spectral pro
Page 227 and 228: 212 J. CuntzThe operator s 1 plays
Page 229 and 230: 214 J. CuntzProposition 7.4. The K-
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258 U. Bunke, T. Schick, M. Spitzwe
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Deformations of gerbes on smooth ma
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Torsion classes of finite type and
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Parshin’s conjecture revisitedTho
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Parshin’s conjecture revisited 41
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428 C. WeibelGiven (0.4) and axiom
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430 C. WeibelAs pointed out in the
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432 C. WeibelConsider the cohomolog
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434 C. WeibelWe need to see that th
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List of contributorsArthur Bartels,
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List of participantsNatalia Abad Sa
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