Boundary and rigidity of nonsingular Bernoulli actions

TL;DR

This paper establishes boundary and rigidity properties for nonsingular Bernoulli actions of exact groups, extending known results from measure-preserving cases to non measure-preserving scenarios using operator algebra techniques.

Contribution

It introduces a boundary construction for Bernoulli crossed product C*-algebras in the nonsingular setting, proving solidity and rigidity results that were previously known only for measure-preserving actions.

Findings

Constructed a boundary for Bernoulli crossed product C*-algebras with certain amenability properties.

Proved that such Bernoulli actions are solid, extending previous measure-preserving results.

First rigidity result for nonsingular Bernoulli actions with two base points.

Abstract

Let $ G $ be a countable discrete group and consider a nonsingular Bernoulli shift action $ G \curvearrowright \prod_{g\in G }(\{0,1\},\mu_g)$ with two base points. When $ G $ is exact, under a certain finiteness assumption on the measures $\{\mu_g\}_{g\in G }$, we construct a boundary for the Bernoulli crossed product C$^*$-algebra that admits some commutativity and amenability in the sense of Ozawa's bi-exactness. As a consequence, we obtain that any such Bernoulli action is solid. This generalizes solidity of measure preserving Bernoulli actions by Ozawa and Chifan--Ioana, and is the first rigidity result in the non measure preserving case. For the proof, we use anti-symmetric Fock spaces and left creation operators to construct the boundary and therefore the assumption of having two base points is crucial.