On operator amenability of Fourier-Stieltjes algebras

TL;DR

This paper investigates the conditions under which the Fourier-Stieltjes algebra B(G) of a locally compact group G is operator amenable, linking it to the structure of G's semitopological compactification and compactness in the connected case.

Contribution

It establishes that operator amenability of B(G) constrains the semitopological compactification of G and implies G's compactness if G is connected.

Findings

Operator amenability of B(G) implies finitely many idempotents in a certain compactification.

For connected G, operator amenability of B(G) implies G is compact.

The study connects algebraic properties of B(G) with topological properties of G.

Abstract

We consider the Fourier-Stietljes algebra B(G) of a locally compact group G. We show that operator amenablility of B(G) implies that a certain semitolpological compactification of G admits only finitely many idempotents. In the case that G is connected, we show that operator amenability of B(G) entails that $G$ is compact.