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

Abstract

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.