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 = S p Lp. Let G be a group and Vcyc the family of virtually cyclic subgroups. Guoliang Yu pro...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2013 |
| País: | Argentina |
| Institución: | Consejo Nacional de Investigaciones Científicas y Técnicas |
| Repositorio: | CONICET Digital (CONICET) |
| Idioma: | inglés |
| OAI Identifier: | oai:ri.conicet.gov.ar:11336/18825 |
| Acceso en línea: | http://hdl.handle.net/11336/18825 |
| Access Level: | acceso abierto |
| Palabra clave: | Algebraic K-theory Schatten ideals Isomorphism conjecture Cyclic homology https://purl.org/becyt/ford/1.1 https://purl.org/becyt/ford/1 |
| Sumario: | 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 = 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 ∗ (E(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 ∗ (E(G, Fin), KH(L p )) ⊗ Q → KH∗(L p [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. |
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