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

We prove the $K$-theoretic Farrell-Jones conjecture for groups as in the title with coefficient rings and $C^{\ast }$-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 $C^{\ast }$-crossed products of $G$ with a stable $C^{\ast }$-algebra have the same $K$-theory.