Properly proximal groups and their von Neumann algebras
Abstract
We introduce a wide class of countable groups, called properly proximal, which contains all non-amenable bi-exact groups, all non-elementary convergence groups, and all lattices in non-compact semi-simple Lie groups, but excludes all inner amenable groups. We show that crossed product II factors arising from free ergodic probability measure preserving actions of groups in this class have at most one weakly compact Cartan subalgebra, up to unitary conjugacy. As an application, we obtain the first -strong rigidity results for compact actions of for .
Keywords
Cite
@article{arxiv.1809.01881,
title = {Properly proximal groups and their von Neumann algebras},
author = {Rémi Boutonnet and Adrian Ioana and Jesse Peterson},
journal= {arXiv preprint arXiv:1809.01881},
year = {2018}
}
Comments
v2, 33 pages. Besides minor changes, we added Theorem 4.3, which provides a more canonical characterization of proper proximality. We use it to simplify the proofs of our main results. This theorem was communicated to us by Narutaka Ozawa