Amenability properties of Rajchman algebras

TL;DR

This paper characterizes when the Rajchman algebra of a locally compact group is amenable, showing it occurs only for compact, almost Abelian groups, and provides examples where it is not approximately unital.

Contribution

It provides a complete characterization of amenability of Rajchman algebras for locally compact groups, extending previous work to non-Abelian cases.

Findings

$B_0(G)$ is amenable iff $G$ is compact and almost Abelian.

Many groups have $B_0(G)$ without an approximate identity.

Examples include non-compact Abelian and infinite solvable groups.

Abstract

Rajchman measures of locally compact Abelian groups are studied for almost a century now, and they play an important role in the study of trigonometric series. Eymard's influential work allowed generalizing these measures to the case of \emph{non-Abelian} locally compact groups $G$. The Rajchman algebra of $G$, which we denote by $B_0(G)$, is the set of all elements of the Fourier-Stieltjes algebra that vanish at infinity. In the present article, we characterize the locally compact groups that have amenable Rajchman algebras. We show that $B_0(G)$ is amenable if and only if $G$ is compact and almost Abelian. On the other extreme, we present many examples of locally compact groups, such as non-compact Abelian groups and infinite solvable groups, for which $B_0(G)$ fails to even have an approximate identity.