Boundary and rigidity of nonsingular Bernoulli actions

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${}^{\ast }$-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.

31 pages