Weak equivalence to Bernoulli shifts for some algebraic actions

TL;DR

This paper demonstrates that certain algebraic actions of countable groups are weakly equivalent to Bernoulli shifts when specific invertibility conditions are met, expanding understanding of their dynamical properties.

Contribution

It establishes weak equivalence to Bernoulli actions for algebraic actions where the defining element is invertible on  but not in the integral group ring, under a formal inverse condition.

Findings

Weak equivalence shown for actions with -invertible but not integral invertible elements.

Extends the class of algebraic actions known to be weakly Bernoulli.

Provides conditions under which algebraic actions mimic Bernoulli shifts.

Abstract

Namely, we prove that if $G$ is a countable, discrete group and $f\in M_{n}(\Z(G))$ is invertible on $\ell^{2}(G)^{\oplus n},$ but $f$ is not invertible in $M_{n}(\Z(G))$, 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 $\ell^{2}$".