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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todo:paper_00029939_v142_n4_p1089_Cortinas2023-10-03T13:55:16Z Operator ideals and assembly maps in K-theory Cortiñas, G. Tartaglia, G. 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. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar 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 |
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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. |
| format |
JOUR |
| author |
Cortiñas, G. Tartaglia, G. |
| spellingShingle |
Cortiñas, G. Tartaglia, G. Operator ideals and assembly maps in K-theory |
| author_facet |
Cortiñas, G. Tartaglia, G. |
| author_sort |
Cortiñas, G. |
| 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 |
| url |
http://hdl.handle.net/20.500.12110/paper_00029939_v142_n4_p1089_Cortinas |
| work_keys_str_mv |
AT cortinasg operatoridealsandassemblymapsinktheory AT tartagliag operatoridealsandassemblymapsinktheory |
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1807320954945142784 |