Operator ultra-amenability

TL;DR

This paper introduces the concept of operator ultra-amenability for Banach algebras and shows that for group algebras, this property implies the group is finite, discrete, and amenable with no infinite abelian subgroups.

Contribution

It extends the notion of ultra-amenability to the operator space setting and characterizes groups with this property as finite, discrete, and highly restricted.

Findings

Operator ultra-amenability implies the group is discrete and amenable.

Such groups have no infinite abelian subgroups.

For certain classes, the group must be finite.

Abstract

Extending M.\ Daws' definition of ultra-amenable Banach algebras, we introduce the notion of operator ultra-amenability for completely contractive Banach algebras. For a locally compact group $G$, we show that the operator ultra-amenability of $A(G)$ imposes severe restrictions on $G$. In particular, it forces $G$ to be a discrete, amenable group with no infinite abelian subgroups. For various classes of such groups, this means that $G$ is finite.