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

Abstract

We prove the K-theoretic Farrell-Jones conjecture for groups with the Haagerup approximation property and coefficient rings and C ∗ C^{∗} -algebras which are stable with respect to compact operators. We use this and Higson-Kasparov's result that the Baum-Connes conjecture holds for such a group G, to show that the algebraic and the C ∗ C^{∗} -crossed product of G with a stable separable G- C ∗ C^{∗} -algebra have the same K-theory.