The Fourier-Stieltjes algebra of a C*-dynamical system

TL;DR

This paper develops a Fourier-Stieltjes algebra for unital discrete twisted C*-dynamical systems, embedding it into the algebra of completely bounded multipliers and characterizing it via positive definiteness and C*-correspondences.

Contribution

It introduces a new Fourier-Stieltjes algebra for C*-dynamical systems, extending classical concepts and providing intrinsic characterizations and embeddings.

Findings

The Fourier-Stieltjes algebra embeds into the algebra of completely bounded multipliers.

A Gelfand-Raikov type theorem characterizes the algebra intrinsically.

The algebra is described via C*-correspondences over crossed products.

Abstract

In analogy with the Fourier-Stieltjes algebra of a group, we associate to a unital discrete twisted C*-dynamical system a Banach algebra whose elements are coefficients of equivariants representations of the system. Building upon our previous work, we show that this Fourier-Stieltjes algebra embeds continuously in the Banach algebra of completely bounded multipliers of the (reduced or full) C*-crossed product of the system. We also introduce a notion of positive definiteness and prove a Gelfand-Raikov type theorem allowing us to describe the Fourier-Stieltjes algebra of a system in a more intrinsic way. After a study of some of its natural commutative subalgebras, we end with a characterization of the Fourier-Stieltjes algebra involving C*-correspondences over the (reduced or full) C*-crossed product.