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: | , |
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| Formato: | JOUR |
| Acceso en línea: | http://hdl.handle.net/20.500.12110/paper_00029939_v142_n4_p1089_Cortinas |
| Aporte de: |
| 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 = ∪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. |
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