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 C * -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) C * -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 C * -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 C * -correspondences over the (reduced or full) C * -crossed product.MSC 2010: 46L55, 43A50, 43A55.

Fourier theory andC∗-algebras

“…To stress some relationship with other notions, the pair (v, ad ρ (σ )) is a kind of (α, σ )-compatible action of G on X that, when σ = 1, gives back a so-called α-α compatible action. There is also a close connection between equivariant representations of Σ and C * -correspondences over C * r (Σ) that can be further exploited [8].…”

“…In analogy with the Fourier-Stieltjes algebra B(G) of a group G, that may be described as the algebra of matrix coefficients of unitary representations of G, one may associate to a unital, discrete, twisted C * -dynamical system Σ = (A, G, α, σ ) its Fourier-Stieltjes algebra B(Σ). We give here a short introduction to this subject, based on [8].…”

“…The presentation covers more topics than strictly needed, with the purpose of providing a reference and a source of inspiration also for future works. In Sections 2 and 3, more biased towards our own contributions [4][5][6][7][8], we present some results illustrating various aspects of Fourier theory in (possibly twisted) discrete reduced group C * -algebras and reduced C * -crossed products.…”

Weak amenability for dynamical systems

“…, is a completely bounded reduced multiplier of (A, G, α), in the sense of Bédos-Conti. The same authors have also introduced a version of the Fourier-Stieltjes algebra for discrete (twisted) C * -dynamical systems, again using Hilbert C * -modules [2].…”

Positive Herz–Schur multipliers and approximation properties of crossed products

“…is a positive element of M n (A). This definition was given by Bédos and Conti [BéC,Definition 4.7] in the more general case of a twisted C * -dynamical system; here we consider only the trivial twist and have simplified the definition accordingly.…”