Trace class operators, regulators, and assembly maps in K-theory

Abstract

Let G be a group, Fin the family of its finite subgroups, and E(G,Fin) the classifying space. Let L^1 be the algebra of trace-class operators in an infinite dimensional, separable Hilbert space over the complex numbers. Consider the rational assembly map in homotopy algebraic K-theory H_p^G(E(G,Fin),KH(L^1))\otimes\Q \to KH_p(L^1[G])\otimes\Q. The rational KH-isomorphism conjecture predicts that the map above is an isomorphism; it follows from a theorem of Yu (see arXiv:1106.3796, arXiv:1202.4999) that it is always injective. In the current article we prove the following. Theorem: Assume that the map above is surjective. Let n\equiv p+1\mod 2. Then: i) The assembly map for the trivial family H_n^G(E(G,{1}),K(\Z)) \to K_n(\Z[G]) is rationally injective. ii) For every number field F, the assembly map H_n^G(E(G,Fin),K(F)) \to K_n(F[G]) is rationally injective. We remark that the K-theory Novikov conjecture asserts that part i) of the theorem above holds for all G, and that part ii) is equivalent to the rational injectivity part of the K-theory Farrell-Jones conjecture for number fields. The idea of the proof of the Theorem is to use an algebraic, equivariant version of Karoubi's multiplicative K-theory, which we introduce in this article.