K-amenability of HNN extensions of amenable discrete quantum groups

We construct the HNN extension of discrete quantum groups, we study their representation theory and we show that an HNN extension of amenable discrete quantum groups is K-amenable

Amenable, transitive and faithful actions of groups acting on trees

We study under which condition an amalgamated free product or an HNN-extension over a finite subgroup admits an amenable, transitive and faithful action on an infinite countable set. We show that such an action exists if the initial groups admit an amenable and almost free action with infinite orbits (e.g. virtually free groups or infinite amenable groups). Our result relies on the Baire category Theorem. We extend the result to groups acting on trees.Comment: v.2: minor changes, final version, to appear in Annales de l'Institut Fourie

On locally compact quantum groups whose algebras are factors

In this paper we are interested in examples of locally compact quantum groups $(M,\Delta)$ such that both von Neumann algebras, $M$ and the dual $\hat{M}$, are factors. There is a lot of known examples such that $(M,\hat{M})$ are respectively of type $(\rm{I}\_{\infty},\rm{I}\_{\infty})$ but there is no examples with factors of other types. We construct new examples of type $(\rm{I}\_{\infty},\rm{II}\_{\infty})$, $(\rm{II}\_{\infty},\rm{II}\_{\infty})$ and $(\rm{III}\_{\lambda},\rm{III}\_{\lambda})$ for each $\lambda\in [0,1]$. Also we show that there is no such example with $M$ or $\hat{M}$ a finite factor.Comment: 20 page

A note on the von Neumann algebra of a Baumslag-Solitar group

We study qualitative properties of the group von Neumann algebra of a Baumslag-Solitar group. Namely, we prove that, in the non-amenable and {ICC} case, the associated ${\rm II}_1$ factor is prime, not solid, and does not have any Cartan subalgebra

The KK-theory of amalgamated free products

We prove a long exact sequence in KK-theory for both full and reduced amalgamated free products in the presence of conditional expectations. In the course of the proof, we established the KK-equivalence between the full amalgamated free product of two unital C*-algebras and a newly defined reduced amalgamated free product that is valid even for non GNS-faithful conditional expectations. Our results unify, simplify and generalize all the previous results obtained before by Cuntz, Germain and Thomsen.Comment: V.3, the paper has been splitted into two papers, this is the first part on amalgamated free product

A locally compact quantum group of triangular matrices

We construct a one parameter deformation of the group of $2\times 2$ upper triangular matrices with determinant 1 using the twisting construction. An interesting feature of this new example of a locally compact quantum group is that the Haar measure is deformed in a non-trivial way. Also, we give a complete description of the dual \cs-algebra and the dual comultiplication

The free wreath product of a compact quantum group by a quantum automorphism group

Let $\mathbb{G}$ be a compact quantum group and $\mathbb{G}^{aut}(B,\psi)$ be the quantum automorphism group of a finite dimensional C*-algebra $(B,\psi)$. In this paper, we study the free wreath product $\mathbb{G}\wr_{*} \mathbb{G}^{aut}(B,\psi)$. First of all, we describe its space of intertwiners and find its fusion semiring. Then, we prove some stability properties of the free wreath product operation. In particular, we find under which conditions two free wreath products are monoidally equivalent or have isomorphic fusion semirings. We also establish some analytic and algebraic properties of $\mathbb{G}\wr_{*} \mathbb{G}^{aut}(B,\psi)$. As a last result, we prove that the free wreath product of two quantum automorphism groups can be seen as the quotient of a suitable quantum automorphism group.Comment: 40 pages, version 2: some statements improved, general presentation improve