Operator ideals and assembly maps in K-theory

Let B be the ring of bounded operators in a complex, separable Hilbert space. For p > 0 consider the Schatten ideal Lp consisting of those operators whose sequence of singular values is p-summable; put S = ∪p Lp. Let G be a group and Vcyc the family of virtually cyclic subgroups. Guoliang Yu...

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Autores principales: Cortiñas, Guillermo Horacio, Tartaglia, Gisela
Publicado: 2014
Acceso en línea:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00029939_v142_n4_p1089_Cortinas
http://hdl.handle.net/20.500.12110/paper_00029939_v142_n4_p1089_Cortinas
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Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
id paper:paper_00029939_v142_n4_p1089_Cortinas
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spelling paper:paper_00029939_v142_n4_p1089_Cortinas2023-06-08T14:23:34Z Operator ideals and assembly maps in K-theory Cortiñas, Guillermo Horacio Tartaglia, Gisela Let B be the ring of bounded operators in a complex, separable Hilbert space. For p > 0 consider the Schatten ideal Lp consisting of those operators whose sequence of singular values is p-summable; put S = ∪p Lp. Let G be a group and Vcyc the family of virtually cyclic subgroups. Guoliang Yu proved that the K-theory assembly map HG * (ε(G, Vcyc),K(S)) → K*(S[G]) is rationally injective. His proof involves the construction of a certain Chern character tailored to work with coefficients 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 HG * (ε(G,Fin),KH(Lp)) ⊗ ℚ → KH*(Lp[G]) ⊗ 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. © 2014 American Mathematical Society. Fil:Cortiñas, G. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Fil:Tartaglia, G. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. 2014 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00029939_v142_n4_p1089_Cortinas http://hdl.handle.net/20.500.12110/paper_00029939_v142_n4_p1089_Cortinas
institution Universidad de Buenos Aires
institution_str I-28
repository_str R-134
collection Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
description Let B be the ring of bounded operators in a complex, separable Hilbert space. For p > 0 consider the Schatten ideal Lp consisting of those operators whose sequence of singular values is p-summable; put S = ∪p Lp. Let G be a group and Vcyc the family of virtually cyclic subgroups. Guoliang Yu proved that the K-theory assembly map HG * (ε(G, Vcyc),K(S)) → K*(S[G]) is rationally injective. His proof involves the construction of a certain Chern character tailored to work with coefficients 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 HG * (ε(G,Fin),KH(Lp)) ⊗ ℚ → KH*(Lp[G]) ⊗ 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. © 2014 American Mathematical Society.
author Cortiñas, Guillermo Horacio
Tartaglia, Gisela
spellingShingle Cortiñas, Guillermo Horacio
Tartaglia, Gisela
Operator ideals and assembly maps in K-theory
author_facet Cortiñas, Guillermo Horacio
Tartaglia, Gisela
author_sort Cortiñas, Guillermo Horacio
title Operator ideals and assembly maps in K-theory
title_short Operator ideals and assembly maps in K-theory
title_full Operator ideals and assembly maps in K-theory
title_fullStr Operator ideals and assembly maps in K-theory
title_full_unstemmed Operator ideals and assembly maps in K-theory
title_sort operator ideals and assembly maps in k-theory
publishDate 2014
url https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00029939_v142_n4_p1089_Cortinas
http://hdl.handle.net/20.500.12110/paper_00029939_v142_n4_p1089_Cortinas
work_keys_str_mv AT cortinasguillermohoracio operatoridealsandassemblymapsinktheory
AT tartagliagisela operatoridealsandassemblymapsinktheory
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