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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Operator ideals and assembly maps in $K$-theory
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by Guillermo Cortiñas and Gisela Tartaglia PDF
Proc. Amer. Math. Soc. 142 (2014), 1089-1099 Request permission

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 \[ H_*^G(\mathcal {E}(G,\mathcal {V}cyc),K(\mathcal {S}))\to K_*(\mathcal {S}[G]) \] 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 \[ H_*^G(\mathcal {E}(G,\mathcal {F}in),KH(\mathcal {L}^p))\otimes \mathbb {Q}\to KH_*(\mathcal {L} ^p[G])\otimes \mathbb {Q} \] 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
  • MR Author ID: 18832
  • ORCID: 0000-0002-8103-1831
  • 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
  • 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
  • © Copyright 2013 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • 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
  • MathSciNet review: 3162232