Weak equivalence to Bernoulli shifts for some algebraic actions

Abstract:

Let $G$ be a countable, discrete group and $f\in M_{n}(\mathbb {Z}(G)).$ We continue our study of the connections between operator theoretic properties of $f$ as a convolution operator on $\ell ^{2}(G)^{\oplus n}$ and the ergodic theoretic properties of the action of $G$ on the Pontryagin dual of $(\mathbb {Z}(G)^{\oplus n}/\mathbb {Z}(G)^{\oplus n}f)$ (denoted $X_{f}$). 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. This explains some of the “Bernoulli-like” properties that $G\curvearrowright X_{f}$ has. We shall in fact prove this weak equivalence in the case that $f$ has a “formal inverse in $\ell ^{2}$”.