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

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

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

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.Comment: 46 pages. Minor revision. A few typos were corrected. The proof that amenability of a system implies its regularity has been streamlined. To appear in Internat. J. Mat

Primitivity of some full group C∗^*-algebras

We show that the full group C$^*$-algebra of the free product of two nontrivial countable amenable discrete groups, where at least one of them has more than two elements, is primitive. We also show that in many cases, this C$^*$-algebra is antiliminary and has an uncountable family of pairwise inequivalent, faithful irreducible representations.Comment: 18 pages. Preliminary version. Comments are wellcome

Fourier theory and C*-algebras

We discuss a number of results concerning the Fourier series of elements in reduced twisted group C∗-algebras of discrete groups, and, more generally, in reduced crossed products associated to twisted actions of discrete groups on unital C∗-algebras. A major part of the article gives a review of our previous work on this topic, but some new results are also included

On maximal ideals in certain reduced twisted C*-crossed products

We consider a twisted action of a discrete group G on a unital C*-algebra A and give conditions ensuring that there is a bijective correspondence between the maximal invariant ideals of A and the maximal ideals in the associated reduced C*-crossed product.Comment: 26 page

Heat properties for groups

We revisit Fourier's approach to solve the heat equation on the circle in the context of (twisted) reduced group C*-algebras, convergence of Fourier series and semigroups associated to negative definite functions. We introduce some heat properties for countably infinite groups and investigate when they are satisfied. Kazhdan's property (T) is an obstruction to the weakest property, and our findings leave open the possibility that this might be the only one. On the other hand, many groups with the Haagerup property satisfy the strongest version. We show that this heat property implies that the associated heat problem has a unique solution regardless of the choice of the initial datum.Comment: 34 pages. Comments welcom

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

We continue our study of the Fourier-Stieltjes algebra associated to a twisted (unital, discrete) C*-dynamical system and discuss how the various notions of equivalence of such systems are reflected at the algebra-level. As an application, we show that the amenability of a system, as defined in our previous work, is preserved under Morita equivalence.Comment: 22 pages. Updated version, in accordance with the comments from the referees. To appear in Studia Mat