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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2014
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| LEADER | 05991caa a22004817a 4500 | ||
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| 008 | 190411s2014 xx ||||fo|||| 00| 0 eng|d | ||
| 024 | 7 | |2 scopus |a 2-s2.0-84893287031 | |
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| 100 | 1 | |a Cortiñas, G. | |
| 245 | 1 | 0 | |a Operator ideals and assembly maps in K-theory |
| 260 | |c 2014 | ||
| 270 | 1 | 0 | |m Departamento de Matemática-IMAS, FCEyN-UBA, Ciudad Universitaria, 1428 Buenos Aires, Argentina |
| 506 | |2 openaire |e Política editorial | ||
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| 520 | 3 | |a 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. |l eng | |
| 593 | |a Departamento de Matemática-IMAS, FCEyN-UBA, Ciudad Universitaria, 1428 Buenos Aires, Argentina | ||
| 700 | 1 | |a Tartaglia, G. | |
| 773 | 0 | |d 2014 |g v. 142 |h pp. 1089-1099 |k n. 4 |p Proc. Am. Math. Soc. |x 00029939 |w (AR-BaUEN)CENRE-347 |t Proceedings of the American Mathematical Society | |
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