Some extremely amenable groups related to operator algebras and ergodic theory

TL;DR

This paper explores the concept of extreme amenability in various groups related to operator algebras and ergodic theory, establishing new characterizations and extending classical results in the field.

Contribution

It provides new characterizations of approximate finite-dimensionality and nuclearity of algebras via extreme amenability of their unitary groups, and proves extreme amenability for several transformation and automorphism groups.

Findings

Unitary group of an AF von Neumann algebra is a product of an extremely amenable and a compact group.

Nuclearity of a C*-algebra is characterized by the strong amenability of its unitary group.

Automorphism groups of Lebesgue spaces and isometries of L^p spaces are extremely amenable.

Abstract

A topological group $G$ is called extremely amenable if every continuous action of $G$ on a compact space has a fixed point. This concept is linked with geometry of high dimensions (concentration of measure). We show that a von Neumann algebra is approximately finite-dimensional if and only if its unitary group with the strong topology is the product of an extremely amenable group with a compact group, which strengthens a result by de la Harpe. As a consequence, a $C^\ast$-algebra $A$ is nuclear if and only if the unitary group $U(A)$ with the relative weak topology is strongly amenable in the sense of Glasner. We prove that the group of automorphisms of a Lebesgue space with a non-atomic measure is extremely amenable with the weak topology and establish a similar result for groups of non-singular transformations. As a consequence, we prove extreme amenability of the groups of isometries of $L^p(0,1)$, $1\leq p<\infty$, extending a classical result of Gromov and Milman ($p=2$). We show that a measure class preserving equivalence relation $\mathcal R$ on a standard Borel space is amenable if and only if the full group $[{\mathcal R}]$, equipped with the uniform topology, is extremely amenable. Finally, we give natural examples of concentration to a nontrivial space in the sense of Gromov occuring in the automorphism groups of injective factors of type $III$.