Actions of dense subgroups of compact groups and ${\text{II}}_1$-factors with the Haagerup property

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 $\mathrm{\Gamma }$ is a countable subgroup of $G$ which has the Haagerup property, then the crossed product algebra $M⋊\mathrm{\Gamma }$ has also 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 products von Neumann algebras are ${\text{II}}_1$-factors with the Haagerup property. If moreover the actions have Property $(\tau )$, then the latter factors are full.