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Equivariant cyclic homology for quantum groups 159Proposition 3.7. Let H be a bornological quantum group and let A be an H -algebra.Then the map A W A Ì H Ì yH ! A y˝ K H is an equivariant algebra isomorphism.For algebraic quantum groups a discussion of Takesaki–Takai duality is containedin [11]. More information on similar duality results in the context of Hopf algebras canbe found in [21].If H D D.G/ is the smooth convolution algebra of a locally compact group G thenan H -algebra is the same thing as a G-algebra. As a special case of Proposition 3.7 oneobtains that for every G-algebra A the double crossed product AÌH Ì yH is isomorphicto the G-algebra A y˝ K G used in [25].4 Radford’s formulaIn this section we prove a formula for the fourth power of the antipode in terms of themodular elements of a bornological quantum group and its dual. This formula wasobtained by Radford in the setting of finite dimensional Hopf algebras [23].Let H be a bornological quantum group. If is a left Haar functional on H thereexists a unique multiplier ı 2 M.H/ such that. y˝ id/.x/ D .x/ıfor all x 2 H . The multiplier ı is called the modular element of H and measuresthe failure of from being right invariant. It is shown in [27] that ı is invertible withinverse S.ı/ D S 1 .ı/ D ı 1 and that one has .ı/ D ı ˝ ı as well as .ı/ D 1.In terms of the dual quantum group the modular element ı defines a character, that is,an essential homomorphism from yH to C. Similarly, there exists a unique modularelement O ı 2 M. yH/for the dual quantum group which satisfies. O y˝ id/ O.f / D O.f / O ıfor all f 2 yH .The Haar functionals of a bornological quantum group are uniquely determined upto a scalar multiple. In many situations it is convenient to fix a normalization at somepoint. However, in the discussion below it is not necessary to keep track of the scalingof the Haar functionals. If ! and are linear functionals we shall write ! if thereexists a nonzero scalar such that ! D . We use the same notation for elements in abornological quantum group or linear maps that differ by some nonzero scalar multiple.Moreover we shall identify H with its double dual using Pontrjagin duality.To begin with observe that the bounded linear functional ı*on H defined byis faithful and satisfies.ı * /.x/ D .xı/..ı * / y˝ id/.x/ D . y˝ id/..xı/.1 ˝ ı 1 // D .xı/ıı 1 D .ı * /.x/:
160 C. VoigtIt follows that ı*is a right Haar functional on H . In a similar way we obtain aright Haar functional (ıon H . Henceı* (ıby uniqueness of the right Haar functional which yields in particular the relationsF l .xı/ G l .x/;F r .ıx/ G r .x/for the Fourier transform.According to Pontrjagin duality we have x D yG l F l S.x/ for all x 2 H . SinceyG l .f / yF l .f O ı/ for every f 2 yH as well as.F l .S.x// ı/.h/ O D .h .1/ S.x// ı.h O .2/ / D .hS.x .2/ // ı.ıS O 2 .x .1/ // F l .S.x .2/ //.h/ ı.x O .1/ / D F l .S.x ( ı//.h/ Owe obtain x yF l F l S.x ( O ı/ or equivalentlyS 1 . O ı*x/ yF l F l .x/: (4.1)Using equation (4.1) and the formula x S 1 yF l F r .x/ obtained from Pontrjaginduality we computewhich impliesyF l F l . O ı 1 *S 2 .x// S 1 S 2 .x/ D S.x/ yF l F r .x/F l .S 2 .x// F r . O ı*x/ (4.2)since yF l is an isomorphism. Similarly, we have yF l SG l id and using yF l .f / yG l .f ı O 1 / together with.SG l .x/ ı O 1 /.h/ D .S.h .1/ /x/ ı O 1 .h .2/ / .S.h/x .2/ / ı O 1 .x .1/ / D SG l .x ( ı O 1 /.h/we obtain yG l SG l .x/ x( O ı. According to the relation F l .xı/ G l .x/ this may berewritten as S 1 yF r F l .xı/ x( O ı which in turn yieldsyF r F l .x/ S..xı 1 /( O ı/: (4.3)Due to Pontrjagin duality we have x D yF r G r S.x/ for all x 2 H and using G r .x/ F r .ıx/ we obtainyF r F r .S.x// yF r G r .ı 1 S.x// D xı: (4.4)According to equation (4.3) and equation (4.4) we haveyF r F l .S 2 .ı 1 .x ( ı O 1 /ı// S..S 2 .ı 1 .x ( ı O 1 /ı/ı 1 /( ı/ O S.S 2 .ı 1 x// D S 1 .ı 1 x/ yF r F r .x/
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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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Page 139 and 140: 124 M. Karoubithe class ˛ of A in
Page 141 and 142: 126 M. Karoubiested in the complex
Page 143 and 144: 128 M. KaroubiThe same method may b
Page 145 and 146: 130 M. KaroubiBy the usual excision
Page 147 and 148: 132 M. KaroubiLet A be a graded twi
Page 149 and 150: 134 M. Karoubithe same ideas as in
Page 151 and 152: 136 M. KaroubiTheorem 6.2. 25 The (
Page 153 and 154: 138 M. KaroubiIn order to show this
Page 155 and 156: 140 M. KaroubiThe following theorem
Page 157 and 158: 142 M. KaroubiWe would like to poin
Page 159 and 160: 144 M. KaroubiZ=n identified with t
Page 161 and 162: 146 M. Karoubiwhere n is the ring
Page 163 and 164: 148 M. Karoubi[6] M. F. Atiyah, R.
Page 166 and 167: 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,
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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 and 222: 206 J. Cuntzprime numbers p and q,
Page 223 and 224: 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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Torsion classes of finite type and
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Parshin’s conjecture revisitedTho
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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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