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.

arXiv commentv2: changed references to and comments on examples at the end of introduction; more general version of Proposition 4.4

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