Ozawa's class $\mathcal S$ for locally compact groups and unique prime factorization

Abstract

We study class $\mathcal S$ for locally compact groups. We characterize locally compact groups in this class as groups having an amenable action on a boundary that is small at infinity, generalizing a theorem of Ozawa. Using this characterization, we provide new examples of groups in class $\mathcal S$ and prove unique prime factorization results for group von Neumann algebras of products of locally compact groups in this class. We also prove that class $\mathcal S$ is a measure equivalence invariant.