C*-simplicity of relative profinite completions of generalized Baumslag-Solitar groups

Low-pass filters and representations of the Baumslag Solitar group

We analyze representations of solvable Baumslag Solitar group that admit wavelets and show how such representations can be constructed from a given low-pass filter. We describe the direct integral decomposition for some examples and derive from it a general criterion for the existence of solutions for scaling equations. As another application, we construct a Fourier transform for some Hausdorff measures.

K-theory for the $C^*$-algebras of the solvable Baumslag-Solitar groups

We provide a new computation of the K-theory of the group $C^*$-algebra of the solvable Baumslag-Solitar group $BS(1,n)\;(n\neq 1)$; our computation is based on the Pimsner-Voiculescu 6-terms exact sequence, by viewing $BS(1,n)$ as a semi-direct product $\mathbb{Z}[1/n]\rtimes\mathbb{Z}$. We deduce from it a new proof of the Baum-Connes conjecture with trivial coefficients for $BS(1,n)$.

Phase transitions on the Toeplitz algebras of Baumslag-Solitar semigroups

Spielberg has recently shown that Baumslag-Solitar groups associated to pairs of positive integers are quasi-lattice ordered in the sense of Nica. Thus they have tractable Toeplitz algebras. Each of these algebras carries a natural dynamics. Here we construct the equilibrium states (the KMS states) for these dynamics. For inverse temperatures larger than a critical value, there is a large simplex of KMS states parametrised by probability measures on the unit circle. At the critical value, and under a mild hypothesis, there is a phase transition in which this simplex collapses to a singleton. T…

自相似作用的半群与高阶Baumslag-Solitar半群

Semigroups of self-similar actions and higher rank Baumslag-Solitar semigroups

本文开创高阶Baumslag-Solitar半群及其C*-代数研究，刻画两类极端情形的von Neumann代数因子性与Cartan子代数，并利用自相似高阶图工具推广秩1情形。

A duality approach to representations of Baumslag-Solitar groups

We give an operator theoretic approach to the constructions of multiresolutions as they are used in a number of basis constructions with wavelets, and in Hilbert spaces on fractals. Our approach starts with the following version of the classical Baumslag-Solitar relations $u t = t^2 u$ where $t$ is a unitary operator in a Hilbert space $\mathcal H$ and $u$ is an isometry in $\mathcal H$. There are isometric dilations of this system into a bigger Hilbert space, relevant for wavelets. For a variety of carefully selected dilations, the ``bigger'' Hilbert space may be $L^2(\br)$, and the dilated o…

C*-algebras associated to graphs of groups

To a large class of graphs of groups we associate a C*-algebra universal for generators and relations. We show that this C*-algebra is stably isomorphic to the crossed product induced from the action of the fundamental group of the graph of groups on the boundary of its Bass-Serre tree. We characterise when this action is minimal, and find a sufficient condition under which it is locally contractive. In the case of generalised Baumslag-Solitar graphs of groups (graphs of groups in which every group is infinite cyclic) we also characterise topological freeness of this action. We are then able t…

HNN extensions of quasi-lattice ordered groups and their operator algebras

The Baumslag-Solitar group is an example of an HNN extension. Spielberg showed that it has a natural positive cone, and that it is then a quasi-lattice ordered group in the sense of Nica. We give conditions for an HNN extension of a quasi-lattice ordered group $(G,P)$ to be quasi-lattice ordered. In that case, if $(G,P)$ is amenable as a quasi-lattice ordered group, then so is the HNN extension.

On the Structure of Some Reduced Amalgamated Free Product C*-Algebras

We study some reduced free products of C*-algebras with amalgamations. We give sufficient conditions for the positive cone of the K_0 group to be the largest possible. We also give sufficient conditions for simplicity and uniqueness of trace. We use the later result to give a necessary and sufficient condition for simplicity and uniqueness of trace of the reduced C*-algebras of the Baumslag-Solitar groups BS(m,n).

Bass-Serre树的边界作用及其在C*-代数中的应用

Boundary actions of Bass-Serre Trees and the applications to $C^*$-algebras

本文研究Bass-Serre理论在C*-代数和拓扑动力学中的应用，探讨图的群的基本群对Bass-Serre树及其边界的动作，识别新的C*-简单群，并研究GBS_n群的C*-简单性。

Compact $κ$-deformation and spectral triples

We construct discrete versions of $κ$-Minkowski space related to a certain compactness of the time coordinate. We show that these models fit into the framework of noncommutative geometry in the sense of spectral triples. The dynamical system of the underlying discrete groups (which include some Baumslag--Solitar groups) is heavily used in order to construct \emph{finitely summable} spectral triples. This allows to bypass an obstruction to finite-summability appearing when using the common regular representation. The dimension of these spectral triples is unrelated to the number of coordinates …