Operator ideals and assembly maps in $K$-theory

Abstract

Let \cB be the ring of bounded operators in a complex, separable Hilbert space. For $p>0$ consider the Schatten ideal \cL^p consisting of those operators whose sequence of singular values is $p$-summable; put \cS=\bigcup_p\cL^p. Let $G$ be a group and \vcyc the family of virtually cyclic subgroups. Guoliang Yu proved that the $K$-theory assembly map H_*^G(\cE(G,\vcyc),K(\cS))\to K_*(\cS[G]) is rationally injective. His proof involves the construction of a certain Chern character tailored to work with coefficients \cS 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 H_*^G(\cE(G,\fin),KH(\cL^p))\otimes\Q\to KH_*(\cL^p[G])\otimes\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.Comment: 11 pages. Version accepted for publication in Proc. Amer. Math. So