Amenability properties of Fourier algebras and Fourier-Stieltjes algebras: a survey

TL;DR

This survey explores the amenability properties of Fourier and Fourier-Stieltjes algebras associated with locally compact groups, emphasizing the role of operator space theory and applications to algebraic structures.

Contribution

It provides a comprehensive overview of amenability, weak amenability, and biflatness of A(G) and B(G), highlighting new insights and methods involving operator spaces.

Findings

A(G) and B(G) exhibit various amenability properties depending on the group G.

Operator space theory is crucial for understanding these algebras' properties.

Applications include results on complemented ideals and homomorphisms.

Abstract

Let G be a locally compact group, and let A(G) and B(G) denote its Fourier and Fourier-Stieltjes algebras. These algebras are dual objects of the group and measure algebras, L^1(G) and M(G), in a sense which generalizes the Pontryagin duality theorem on abelian groups. We wish to consider the amenability properties of A(G) and B(G) and compare them to such properties for L^1(G) and M(G). For us, ``amenability properties'' refers to amenability, weak amenability, and biflatness, as well as some properties which are more suited to special settings, such as the hyper-Tauberian property for semisimple commutative Banach algebras. We wish to emphasize that the theory of operator spaces and completely bounded maps plays an indispensable role when studying A(G) and B(G). We also show some applications of amenability theory to problems of complemented ideals and homomorphisms.