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

Abstract

In analogy with the Fourier–Stieltjes algebra of a group, we associate to a unital discrete twisted [Formula: see text]-dynamical system a Banach algebra whose elements are coefficients of equivariant 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) [Formula: see text]-crossed product of the system. We 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. We also propose a definition of amenability for [Formula: see text]-dynamical systems and show that it implies regularity. After a study of some natural commutative subalgebras, we end with a characterization of the Fourier–Stieltjes algebra involving [Formula: see text]-correspondences over the (reduced or full) [Formula: see text]-crossed product.