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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
DOI: https://doi.org/10.1090/S0002-9939-2013-11837-X
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.

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Additional Information

Guillermo Cortiñas
Affiliation: Departamento de Matemática-IMAS, FCEyN-UBA, Ciudad Universitaria, 1428 Buenos Aires, Argentina
Email: gcorti@dm.uba.ar

Gisela Tartaglia
Affiliation: Departamento de Matemática-IMAS, FCEyN-UBA, Ciudad Universitaria, 1428 Buenos Aires, Argentina
Email: gtartag@dm.uba.ar

DOI: https://doi.org/10.1090/S0002-9939-2013-11837-X
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.