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Ozawa's class 𝒮 for locally compact groups and unique prime factorization of group von Neumann algebras

Published online by Cambridge University Press:  19 July 2019

Tobe Deprez*
Department of Mathematics, KU Leuven, Celestijnenlaan 200B, 3001Leuven, Belgium (


We study class 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 𝒮 and prove a unique prime factorization theorem for group von Neumann algebras of products of locally compact groups in this class. We also prove that class 𝒮 is a measure equivalence invariant.

Research Article
Copyright © Royal Society of Edinburgh 2019

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