Actions of dense subgroups of compact groups and II1-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 Gamma is a countable subgroup of G which has the Haagerup property, then the crossed product algebra M x 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 II(1)-factors with the Haagerup property. Moreover, if the actions have property (tau), then the latter factors are full.