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We study C*-irreducibility of inclusions of reduced twisted group C*-algebras and of reduced group C*-algebras. We characterize C*-irreducibility in the case of an inclusion arising from a normal subgroup, and exhibit many new examples of C*-irreducible inclusions., Comment: 23 pages. v2: A couple of minor changes and some typos corrected. The paper has been slightly reorganized. v3: Final version, incorporating some improvements suggested by the referee. To appear in jFA

This paper provides sufficient density conditions for the existence of smooth vectors generating a frame or Riesz sequence in the lattice orbit of a square-integrable projective representation of a nilpotent Lie group. The conditions involve the product of lattice co-volume and formal dimension, and complement Balian-Low type theorems for the non-existence of smooth frames and Riesz sequences at the critical density. The proof hinges on a connection between smooth lattice orbits and generators for an explicitly constructed finitely generated Hilbert $C^*$-module. An important ingredient in the approach is that twisted group $C^*$-algebras associated to finitely generated nilpotent groups have finite decomposition rank, hence finite nuclear dimension, which allows us to deduce that any matrix algebra over such a simple $C^*$-algebra has strict comparison of projections., Comment: 36 pages

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.

In Proposition 2.9 of our paper cited above, we asserted that the space of bounded twisted multipliers on a discrete group is independent of the twist. It appears that this statement is not true in general. Only a weaker statement can be deduced from our arguments, as we explain in this short note. The rest of our article is unaffected by this error, except for one assertion in Corollary 2.12, which we also correct.

When a locally compact group acts on a C ⁎ -correspondence, it also acts on the associated Cuntz–Pimsner algebra in a natural way. Hao and Ng have shown that when the group is amenable the Cuntz–Pimsner algebra of the crossed product correspondence is isomorphic to the crossed product of the Cuntz–Pimsner algebra. In this paper, we have a closer look at this isomorphism in the case where the group is not necessarily amenable. We also consider what happens at the level of Toeplitz algebras.

Given an action $$\alpha $$ of a discrete group G on a unital $$C^*$$ -algebra A, we introduce a natural concept of $$\alpha $$ -negative definiteness for functions from G to A, and examine some of the first consequences of such a notion. In particular, we prove analogs of theorems due to Delorme–Guichardet and Schoenberg in the classical case where A is trivial. We also give a characterization of the Haagerup property for the action $$\alpha $$ when G is countable.

We study the problem of determining when the reduced twisted group C*-algebra associated with a discrete group G is simple and/or has a unique tracial state, and present new sufficient conditions for this to hold. One of our main tools is a combinatorial property, that we call the relative Kleppner condition, which ensures that a quotient group G/H acts by freely acting automorphisms on the twisted group von Neumann algebra associated to a normal subgroup H. We apply our results to different types of groups, e.g. wreath products and Baumslag-Solitar groups., 37 pages. Section 4 has been slightly reorganized and improved, and one example has been added to Section 5. A few redactional changes and some typos corrected throughout the text. Final version, to appear in J. Noncommut. Geom

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.

This paper is an invitation to Fourier analysis in the context of reduced twisted C*-crossed products associated with discrete unital twisted C*-dynamical systems. We discuss norm-convergence of Fourier series, multipliers and summation processes. Our study relies in an essential way on the (covariant and equivariant) representation theory of C*-dynamical systems on Hilbert C*-modules. It also yields some information on the ideal structure of reduced twisted C*-crossed products., Comment: Some misprints corrected, Section 5 has been slightly reorganized

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 lcqg's.

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.

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., 26 pages

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)., Comment: 25 pages. LaTex

Let (α, u) denote a cocycle-crossed action of a discrete group G on a unital C*-algebra A, and B = C*r (A, G, α, u) the associated reduced (twisted) C*-crossed product. We discuss the following problem: when is B a simple C*-algebra with a unique trace? To this aim, we introduce the concept of tracially properly outerness of actions.

Let λ \lambda denote the left regular representation of a locally compact group G G on L 2 ( G ) {L^2}(G) and C ∗ ( λ ( G ) ) {C^{\ast }}(\lambda (G)) the C ∗ {C^{\ast }} -algebra generated by λ ( G ) \lambda (G) . We show that the amenability of G G and the amenability of G G considered as a discrete group both may be characterized in terms of C ∗ ( λ ( G ) ) {C^{\ast }}(\lambda (G)) .

We show that the full group C$^*$-algebra of $PSL(n, \Z)$ is primitive when $n=2$, and not primitive when $n\geq 3$. Moreover, we show that there exists an uncountable family of pairwise inequivalent, faithful irreducible representations of $C^*(PSL(2,\Z))$., Comment: 14 pages. Some changes in the exposition.

We study norm convergence and summability of Fourier series in the setting of reduced twisted group $C^*$-algebras of discrete groups. For amenable groups, F{\o}lner nets give the key to Fej\'er summation. We show that Abel-Poisson summation holds for a large class of groups, including e.g. all Coxeter groups and all Gromov hyperbolic groups. As a tool in our presentation, we introduce notions of polynomial and subexponential H-growth for countable groups w.r.t. proper scale functions, usually chosen as length functions. These coincide with the classical notions of growth in the case of amenable groups., Comment: 35 pages; abridged, revised and updated

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.

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 qroup has a unitary generator which may be described in a concrete way., 34 pages

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).

We show that a reflective/coreflective pair of full subcategories satisfies a "maximal-normal"-type equivalence if and only if it is an associated pair in the sense of Kelly and Lawvere., Comment: added section numbers

We initiate a study of infinite tensor products of projective unitary representations of a discrete group G. Special attention is given to regular representations twisted by 2-cocycles and to projective representations associated with CCR-representations of bilinear maps. Detailed computations are presented in the case where G is a finitely generated free abelian group. We also discuss an extension problem about product type actions of G, where the projective representation theory of G plays a central role., 21 pages, Latex; updated version with few slight changes, some Comments have been deleted. Rocky Mountain J. Math., to appear

We introduce and study several amenability properties for unitary corepresentations and *-representations of algebraic quantum groups, which may be used to characterize amenability or co-amenability of such groups. As a background for this study, we also investigate the involved tensor C*-categories., 46 pages, (some misprints corrected, a couple of remarks and a new reference added)