Weak equivalence to Bernoulli shifts for some algebraic actions

Abstract

Namely, we prove that if $G$ is a countable, discrete group and $f\in M_n(\mathbb{Z}(G))$ is invertible on ${\ell }^2(G{)}^{\oplus n},$ but $f$ is not invertible in $M_n(\mathbb{Z}(G))$, then the measure-preserving action of $G$ on $X_f$ equipped with the Haar measure is weakly equivalent to a Bernoulli action. We shall in fact prove this weak equivalence in the case that $f$ has a "formal inverse in ${\ell }^2$".

12 pages. This is the final version, to appear as such in Proceedings of the American Mathematical Society