Erik Christopher Bedos

Co-Amenability of Compact Quantum Groups

Abstract We study the concept of co-amenability for a compact quantum group. Several conditions are derived that are shown to be equivalent to it. Some consequences of co-amenability that we obtain are faithfulness of the Haar integral and automatic norm-boundedness of positive linear functionals on the quantum group’s Hopf ∗ -algebra (neither of these properties necessarily holds without co-amenability).

Amenability and Co-Amenability for Locally Compact Quantum Groups

We define concepts of amenability and co-amenability for locally compact quantum groups in the sense of J. Kustermans and S. Vaes. Co-amenability of a lcqg (locally compact quantum group) is proved to be equivalent to a series of statements, all of which imply amenability of the dual lcqg. Further, it is shown that if a lcqg is amenable, then its universal dual lcqg is nuclear. We also define and study amenability and weak containment concepts for representations and corepresentations of lcqgs.

On amenability and co-amenability of algebraic quantum groups and their corepresentations

We introduce and study several notions of amenability for unitary corepresentations and ∗-representations of algebraic quantum groups, which may be used to characterize amenability and co-amenability for such quantum groups. As a background forthis study, we investigate the associated tensor C � -categories.

AMENABILITY AND COAMENABILITY OF ALGEBRAIC QUANTUM GROUPS

We define concepts of amenability and coamenability for algebraic quantum groups in the sense of Van Daele (1998). We show that coamenability of an algebraic quantum group always implies amenability of its dual. Various necessary and/or sufficient conditions for amenability or coamenability are obtained. Coamenability is shown to have interesting consequences for the modular theory in the case that the algebraic quantum group is of compact type.

Discrete groups and simple C*-algebras

Let G denote a discrete group and let us say that G is C *-simple if the reduced group C *-algebra associated with G is simple. We notice immediately that there is no interest in considering here the full group C *-algebra associated with G , because it is simple if and only if C is trivial. Since Powers in 1975 [26] proved that all nonabelian free groups are C *-simple, the class of C *-simp1e groups has been considerably enlarged (see [1, 2, 6, 7, 12, 13, 14, 16, 24] as a sample!), and two important subclasses are so-called weak Powers groups ([6, 13]; see Section 4 for definition and examples) and the groups of Akemann-Lee type [1, 2], which are groups possessing a normal non-abelian free subgroup with trivial centralizer.

Amenability and co-amenability of algebraic quantum groups II

Abstract We continue our study of the concepts of amenability and co-amenability for algebraic quantum groups in the sense of A. Van Daele and our investigation of their relationship with nuclearity and injectivity. One major tool for our analysis is that every non-degenerate ∗ -representation of the universal C ∗ -algebra associated to an algebraic quantum group has a unitary generator which may be described in a concrete way.

On actions of amenable groups on II1-factors

Abstract Given a II 1 -factor M with separable predual and α, a free action of a countable amenable discrete group G on M , we show that the crossed product M × α G has property Γ (resp. is McDuff) when M itself has property Γ (resp. is McDuff).

The full group C -algebra of the modular group is primitive

We show that the full group C*-algebra of PSL(n,Z) is primitive when n = 2 and not primitive when n >= 3. Moreover, we show that there exists an uncountable family of pairwise inequivalent, faithful irreducible representations of C*(PSL(2,Z)).

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.

An introduction to 3D discrete magnetic Laplacians and noncommutative 3-tori

Abstract We initiate a study of spectral properties of 3D discrete magnetic Laplacians based on their relationship to noncommutative 3-tori.