@phdthesis{Deprez2019thesis,
author = {Deprez, Tobe},
school = {KU Leuven},
title = {Rigidity for Von Neumann Algebras Given by Locally Compact Groups},
type = {PhD thesis},
year = {2019},
abstract = {Given a group {$G$}, the group von Neumann algebra {$L(G)$} is defined as the {w.o.\ closure} of the linear span of the left regular representation {$\{\lambda_g\}_{g\in G}$}. The group measure space construction of Murray and von Neumann associates to every nonsingular action {$G\acts (X,\mu)$} a von Neumann algebra {$L^\infty(X)\rtimes G$}. A central problem in the study of von Neumann algebras is the classification of {$L(G)$} and {$L^\infty(X)\rtimes G$} in terms of the underlying group {$G$} and the underlying action {$G\acts (X,\mu)$}, respectively. By defining Ozawa's class {$\mathcal S$} for locally compact groups, we obtain the first rigidity and classification results for group von Neumann algebras and group measure space von Neumann algebras given by nondiscrete, locally compact groups.
Class {$\mathcal S$} for discrete groups plays an important role in rigidity results for group von Neumann algebras and group measure space von Neumann algebras given by discrete groups. We define class {$\mathcal S$} for locally compact
groups and 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. We provide examples of locally compact groups in class {$\mathcal S$} and prove that class {$\mathcal S$} is closed under measure equivalence.
We prove that for arbitrary free, probability measure preserving actions of weakly amenable groups in class {$\mathcal S$}, the group measure space von Neumann algebra {$L^\infty(X)\rtimes G$} has a unique Cartan subalgebra up to unitary conjugacy. We then deduce a {\W*-strong} rigidity theorem of irreducible actions of products of such groups. These theorems in particular apply to connected, simple Lie groups of real rank one with finite center and groups acting properly on trees or hyperbolic graphs. We furthermore prove strong solidity results for the group von Neumann algebras of weakly amenable groups in class {$\mathcal S$}, and we prove a unique prime factorization result for group von Neumann algebras of products of groups in class {$\mathcal S$}.}
}