On Fourier and Fourier-Stieltjes Algebras of C$^\ast$-Dynamical Systems

TL;DR

This paper advances the understanding of Fourier and Fourier-Stieltjes algebras associated with C*-dynamical systems, introducing a new Fourier algebra concept and analyzing equivariant representations, especially for commutative systems.

Contribution

It defines a natural Fourier algebra for C*-dynamical systems and characterizes it via multipliers, also analyzing equivariant representations in the commutative case.

Findings

Fourier algebra can be defined as the closure of multipliers with finite support.

Equivariant representation theory relates to cocycle representations of transformation groups.

Main result links group theory with equivariant representations of commutative systems.

Abstract

We continue the study of the Fourier-Stieltjes algebra of a C$^\ast$-dynamical system, initiated by B\'edos and Conti, and recently extended by Buss, Kwa\'sniewski, McKee and Skalski. Firstly, we introduce and study a natural notion of a Fourier algebra of a C$^\ast$-dynamical system. Notably, we show that it can be equivalently defined as the closure of the multipliers with finite support or as the closure of multipliers coming from regular equivariant representations. Secondly, we undertake an analysis of the equivariant representation theory of commutative systems. Our main result about this is a description of the group theoretical aspect of the equivariant representation theory in terms of cocycle representations of the underlying transformation group.