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Operator ideals and assembly maps in $ K$-theory

Authors: Guillermo Cortiñas and Gisela Tartaglia
Journal: Proc. Amer. Math. Soc. 142 (2014), 1089-1099
MSC (2010): Primary 19D50, 19D55, 19K99
Published electronically: December 20, 2013
MathSciNet review: 3162232
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Abstract: Let $ \mathcal {B}$ be the ring of bounded operators in a complex, separable Hilbert space. For $ p>0$ consider the Schatten ideal $ \mathcal {L}^p$ consisting of those operators whose sequence of singular values is $ p$-summable; put $ \mathcal {S}=\bigcup _p\mathcal {L}^p$. Let $ G$ be a group and $ \mathcal {V}cyc$ the family of virtually cyclic subgroups. Guoliang Yu proved that the $ K$-theory assembly map

$\displaystyle H_*^G(\mathcal {E}(G,\mathcal {V}cyc),K(\mathcal {S}))\to K_*(\mathcal {S}[G]) $

is rationally injective. His proof involves the construction of a certain Chern character tailored to work with coefficients $ \mathcal {S}$ and the use of some results about algebraic $ K$-theory of operator ideals and about controlled topology and coarse geometry. In this paper we give a different proof of Yu's result. Our proof uses the usual Chern character to cyclic homology. Like Yu's, it relies on results on algebraic $ K$-theory of operator ideals, but no controlled topology or coarse geometry techniques are used. We formulate the result in terms of homotopy $ K$-theory. We prove that the rational assembly map

$\displaystyle H_*^G(\mathcal {E}(G,\mathcal {F}in),KH(\mathcal {L}^p))\otimes \mathbb{Q}\to KH_*(\mathcal {L} ^p[G])\otimes \mathbb{Q} $

is injective. We show that the latter map is equivalent to the assembly map considered by Yu, and thus obtain his result as a corollary.

References [Enhancements On Off] (What's this?)

  • [1] Arthur Bartels, Tom Farrell, Lowell Jones, and Holger Reich, On the isomorphism conjecture in algebraic $ K$-theory, Topology 43 (2004), no. 1, 157-213. MR 2030590 (2004m:19004),
  • [2] Arthur Bartels and Wolfgang Lück, Isomorphism conjecture for homotopy $ K$-theory and groups acting on trees, J. Pure Appl. Algebra 205 (2006), no. 3, 660-696. MR 2210223 (2007e:19005),
  • [3] Guillermo Cortiñas, The obstruction to excision in $ K$-theory and in cyclic homology, Invent. Math. 164 (2006), no. 1, 143-173. MR 2207785 (2006k:19006),
  • [4] G. Cortiñas and E. Ellis, Isomorphism conjectures with proper coefficients. arXiv:1108.5196
  • [5] Guillermo Cortiñas and Andreas Thom, Comparison between algebraic and topological $ K$-theory of locally convex algebras, Adv. Math. 218 (2008), no. 1, 266-307. MR 2409415 (2009h:46136),
  • [6] Joachim Cuntz, Cyclic theory and the bivariant Chern-Connes character, Noncommutative geometry, Lecture Notes in Math., vol. 1831, Springer, Berlin, 2004, pp. 73-135. MR 2058473 (2005e:58007),
  • [7] Joachim Cuntz and Daniel Quillen, Excision in bivariant periodic cyclic cohomology, Invent. Math. 127 (1997), no. 1, 67-98. MR 1423026 (98g:19003),
  • [8] James F. Davis and Wolfgang Lück, Spaces over a category and assembly maps in isomorphism conjectures in $ K$- and $ L$-theory, $ K$-Theory 15 (1998), no. 3, 201-252. MR 1659969 (99m:55004),
  • [9] Wolfgang Lück and Holger Reich, Detecting $ K$-theory by cyclic homology, Proc. London Math. Soc. (3) 93 (2006), no. 3, 593-634. MR 2266961 (2007h:19008),
  • [10] Randy McCarthy, The cyclic homology of an exact category, J. Pure Appl. Algebra 93 (1994), no. 3, 251-296. MR 1275967 (95b:19002),
  • [11] John Roe, Coarse cohomology and index theory on complete Riemannian manifolds, Mem. Amer. Math. Soc. 104 (1993), no. 497, x+90. MR 1147350 (94a:58193)
  • [12] Andrei A. Suslin and Mariusz Wodzicki, Excision in algebraic $ K$-theory, Ann. of Math. (2) 136 (1992), no. 1, 51-122. MR 1173926 (93i:19006),
  • [13] R. W. Thomason, Algebraic $ K$-theory and étale cohomology, Ann. Sci. École Norm. Sup. (4) 18 (1985), no. 3, 437-552. MR 826102 (87k:14016)
  • [14] Charles A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994. MR 1269324 (95f:18001)
  • [15] Charles A. Weibel, Homotopy algebraic $ K$-theory, Algebraic $ K$-theory and algebraic number theory (Honolulu, HI, 1987), Contemp. Math., vol. 83, Amer. Math. Soc., Providence, RI, 1989, pp. 461-488. MR 991991 (90d:18006),
  • [16] Mariusz Wodzicki, Algebraic $ K$-theory and functional analysis, First European Congress of Mathematics, Vol. II (Paris, 1992) Progr. Math., vol. 120, Birkhäuser, Basel, 1994, pp. 485-496. MR 1341858 (97f:46112)
  • [17] G. Yu, The algebraic K-theory Novikov conjecture for group algebras over the ring of Schatten class operators. arXiv:1106.3796

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Additional Information

Guillermo Cortiñas
Affiliation: Departamento de Matemática-IMAS, FCEyN-UBA, Ciudad Universitaria, 1428 Buenos Aires, Argentina

Gisela Tartaglia
Affiliation: Departamento de Matemática-IMAS, FCEyN-UBA, Ciudad Universitaria, 1428 Buenos Aires, Argentina

Received by editor(s): March 15, 2012
Received by editor(s) in revised form: April 24, 2012
Published electronically: December 20, 2013
Additional Notes: The first author was partially supported by MTM2007-64704
Both authors were supported by CONICET and partially supported by grants UBACyT 20020100100386, PIP 112-200801-00900 and MathAmSud project U11Math05.
Communicated by: Brooke Shipley
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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