arXiv subjects
Publications and source records attributed to Erik Bédos.
We introduce new properties of Haagerup-type for Fell bundles over discrete groups and discuss their relationship with Haagerup properties for the associated $C^*$-algebras. In particular, our results on Fell bundles associated with twisted unital $C^*$-dynamical systems extend previous results known in the untwisted case.
We study methods of inducing metrics on unital completely positive maps by employing seminorms arising in noncommutative geometry. Our main approach relies on the development of an infinite-dimensional $C^*$-algebraic analogue of the Choi-Jamiołkowski isomorphism. Under suitable conditions, we show that the induced metrics satisfy the quantum information theoretic properties of stability and chaining. Moreover, we show how to generate such metrics using constructions native to noncommutative geometry, by for example using external Kasparov products of spectral triples.
We introduce and study actions of Fell bundles over discrete groups on Hilbert bundles. Many examples of such actions are presented. We discuss the connection with positive definite bundle maps between Fell bundles, culminating in the unital case in a Gelfand-Raikov type theorem. We also use these actions to construct C*-correspondences over cross-sectional C*-algebras of Fell bundles.
We present a new method, involving monads and comonads from category theory, to help establish a certain type of equivalence of subcategories. As a case study we consider the category of topological gradings of $C^*$-algebras over a fixed Hausdorff étale groupoid.
We introduce a natural concept of positive definiteness for bundle maps between Fell bundles over (possibly different) discrete groups and describe several examples. Such maps induce completely positive maps between the associated full cross-sectional $C^*$-algebras in a functorial way. Under the assumption that the kernel of the homomorphism connecting the groups under consideration is amenable, they also induce completely positive maps between the associated reduced cross-sectional $C^*$-algebras. As an application, we define an approximation property for a Fell bundle over a discrete group which generalizes Exel's approximation property and still implies the weak containment property. Both approximation properties coincide when the unit fibre is nuclear.
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.
It is well-known that the maximalization of a coaction of a locally compact group on a C*-algebra enjoys a universal property. We show how this important property can be deduced from a categorical framework by exploiting certain properties of the maximalization functor for coactions. We also provide a dual proof for the universal property of normalization of coactions.
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.
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.
Given a group cocycle on a finitely aligned left cancellative small category (LCSC) we investigate the associated skew product category and its Cuntz-Krieger algebra, which we describe as the crossed product of the Cuntz-Krieger algebra of the original category by an induced coaction of the group. We use our results to study Cuntz-Krieger algebras arising from free actions of groups on finitely aligned LCSC's, and to construct coactions of groups on Exel-Pardo algebras. Finally we discuss the universal group of a small category and connectedness of skew product categories.
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.
We consider Toeplitz and Cuntz-Krieger $C^*$-algebras associated with finitely aligned left cancellative small categories. We pay special attention to the case where such a category arises as the Zappa-Szép product of a category and a group linked by a one-cocycle. As our main application, we obtain a new approach to Exel-Pardo algebras in the case of row-finite graphs. We also present some other ways of constructing $C^*$-algebras from left cancellative small categories and discuss their relationship.
We generalize a recent construction of Exel and Pardo, from discrete groups acting on finite directed graphs to locally compact groups acting on topological graphs. To each cocycle for such an action, we construct a $C^*$-correspondence whose associated Cuntz-Pimsner algebra is the analog of the Exel-Pardo $C^*$-algebra.
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.
We prove a version of Wordingham's theorem for left regular representations in the setting of Fell bundles of inverse semigroups and use this result to discuss the various associated cross sectional C*-algebras.
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.
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.
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.