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