Connes-amenability and normal, virtual diagonals for measure algebras, II

TL;DR

This paper establishes the equivalence between group amenability, Connes-amenability of measure algebras, and the existence of a normal, virtual diagonal for measure algebras of locally compact groups.

Contribution

It proves the equivalence of three key properties for locally compact groups, linking group amenability with algebraic and functional-analytic conditions.

Findings

G is amenable if and only if M(G) is Connes-amenable

G is amenable if and only if M(G) has a normal, virtual diagonal

The results unify group properties with measure algebra structures

Abstract

We prove that the following are equivalent for a locally compact group $G$: (i) $G$ is amenable; (ii) $M(G)$ is Connes-amenable; (iii) $M(G)$ has a normal, virtual diagonal.