Compact operators and algebraic $K$-theory for groups which act properly and isometrically on Hilbert space

TL;DR

This paper proves the Farrell-Jones conjecture for a class of groups acting properly and isometrically on Hilbert space, establishing new links between algebraic and operator algebraic $K$-theory.

Contribution

It extends the validity of the Farrell-Jones conjecture to these groups with stable coefficient rings and $C^*$-algebras, connecting algebraic and $C^*$-crossed product $K$-theories.

Findings

Farrell-Jones conjecture holds for these groups.

Algebraic and $C^*$-crossed product $K$-theories coincide for stable $C^*$-algebras.

Uses Higson-Kasparov's result on Baum-Connes conjecture with coefficients.

Abstract

We prove the $K$-theoretic Farrell-Jones conjecture for groups as in the title with coefficient rings and $C^*$-algebras which are stable with respect to compact operators. We use this and Higson-Kasparov's result that the Baum-Connes conjecture with coefficients holds for such groups, to show that if $G$ is as in the title then the algebraic and the $C^*$-crossed products of $G$ with a stable $C^*$-algebra have the same $K$-theory.