Whirliness beyond amenability

Abstract

Given a non-atomic standard probability space $(\Omega ,\mu )$, we exhibit examples of non-amenable closed topological subgroups of $\mathrm{Aut}(\Omega ,\mu )$ whose natural near-action on $(\Omega ,\mu )$ is whirly. This answers a 2010 question by Pestov in the negative. The argument proceeds via constructing Gaussian near-actions from topologically faithful unitary representations of non-amenable whirly Polish groups.

25 pages, no figures