Actions of dense subgroups of compact groups and II$_{1}$-factors with the Haagerup property

Abstract

Let $M$ be a finite von Neumann algebra with the Haagerup property, and let $G$ be a compact group that acts continuously on $M$ and that preserves some finite trace $\tau$. We prove that, if $\Gamma$ is a countable subgroup of $G$ which has the Haagerup property, then the crossed product algebra $M\rtimes\Gamma$ also has the Haagerup property. In particular, we study some ergodic, non-weakly mixing actions of groups with the Haagerup property on finite, injective von Neumann algebras, and we prove that the associated crossed product von Neumann algebras are $\textrm{II}_1$-factors with the Haagerup property. Moreover, if the actions have property $(\tau)$, then the latter factors are full.