Notes on treeability and costs for discrete groupoids in operator algebra framework

关于FLC点集的内顺从性注记

A note on inner amenability for FLC point sets

本文证明第二可数局部紧群中近似格关联的群胚是内顺从的，并将结果推广到有限局部复杂性点集，部分回应了Ananantharman-Delaroche关于étale群胚内顺从性的公开问题。

Étale groupoids and their $C^*$-algebras

These notes were written as supplementary material for a five-hour lecture series presented at the Centre de Recerca Mathemàtica at the Universitat Autònoma de Barcelona from the 13th to the 17th of March 2017. The intention of these notes is to give a brief overview of some key topics in the area of $C^*$-algebras associated to étale groupoids. The scope has been deliberately contained to the case of étale groupoids with the intention that much of the representation-theoretic technology and measure-theoretic analysis required to handle general groupoids can be suppressed in this simpler setti…

All classifiable Kirchberg algebras are $C^{\ast}$-algebras of ample groupoids

In this note we show that every Kirchberg algebra in the UCT class is the $C^{\ast}$-algebra of a Hausdorff, ample, amenable and locally contracting groupoid. The non-unital case follows from Spielberg's graph-based models for Kirchberg algebras. Our contribution is the unital case and our proof builds on Spielberg's construction.

扭群胚C*-代数的局部对合假设

The local bisection hypothesis for twisted groupoid C*-algebras

本文提出了等价于局部紧Hausdorff群胚有效性的判据，特别是C*-代数局部对合假设，并利用正规化子半群在约化扭群胚C*-代数和循环群C*-代数中的基础作用完成证明。

Representing Kirchberg algebras as inverse semigroup crossed products

In this note we show that a combinatorial model of Kirchberg algebras in the UCT, namely the Katsura algebras O_{AB}, can be expressed both as groupoid C*-algebras and as inverse semigroup crossed products. We use this picture to obtain results about simplicity, pure infiniteness and nuclearity of O_{AB} using different methods than those used by Katsura.

A note on the core of Steinberg algebras

For an ample Hausdorff groupoid $G$, and the Steinberg algebra $A_R(G)$ with coefficients in the commutative ring $R$ with unit, we describe the centraliser of subalgebra $A_R(U)$ with $U$ an open closed invariant subset of unit space of $G$. In particular, we obtain that the algebra of the interior of the isotropy is indeed the centraliser of the diagonal subalgebra of Steinberg algebra. This will unify several results in the literature and the corresponding results for Leavitt path algebras follow.

A note on the quasi-diagonality of inverse semigroup reduced C*-algebras

In this note we start the study of whether the reduced C*-algebra of an inverse semigroup is quasi-diagonal, making explicit use of the inner structure of this class of semigroups in order to produce quasi-diagonal approximations. Given a discrete inverse semigroup, we detail the relationship between its isolated subgroups and the quasi-diagonality of its reduced C*-algebra, and prove that such subgroups must be amenable. Moreover, we give a direct characterization of the quasi-diagonality of inverse semigroup whose universal groupoid is minimal. Lastly, we also study the relevance of Green's …

Wodzicki 留数与 Kontsevich-Vishik 迹的群胚方法

A groupoid approach to the Wodzicki residue and the Kontsevich-Vishik trace

本文将 Couchet 和 Yuncken 关于 $-\dim(M)$ 阶伪微分算子 Wodzicki 留数的群胚方法推广到任意阶，并给出 Kontsevich-Vishik 迹的群胚描述。

关于双曲3-流形群边界作用所关联C*-代数的注记

Note on C*-algebras associated to boundary actions of hyperbolic 3-manifold groups

本文利用Kirchberg-Phillips分类证明双曲3-流形群边界交叉积C*-代数的同构类型仅由流形同调决定，并构造出无穷多个不同构但交叉积同构的双曲群，且这些同构不具有动力学来源。

C*-algebras associated with branched coverings

In this note we analyze the C*-algebra associated with a branched covering both as a groupoid C*-algebra and as a Cuntz-Pimsner algebra. We determine conditions when the algebra is simple and purely infinite. We indicate how to compute the K-theory of several examples, including one related to rational maps on the Riemann sphere.