Operator ideals and assembly maps in K-theory

Abstract

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.