Operator Algebras and Mauldin Williams Graphs

étale 群胚上 Fell 丛的 C$^*$-代数

C$^*$-algebras of Fell bundles over étale groupoids

本文在不使用 Renault 分解定理的前提下，构造了非饱和 Fell 丛在非豪斯多夫局部紧 étale 群胚上的全 C$^*$-代数，推广了 Muhly-Williams 标准构造。

Cohomology of topological graphs and Cuntz-Pimsner algebras

We compute the sheaf cohomology of a groupoid built from a local homeomorphism of a locally compact space $X$. In particular, we identify the twists over this groupoid, and its Brauer group. Our calculations refine those made by Kumjian, Muhly, Renault and Williams in the case $X$ is the path space of a graph, and the local homeomorphism is the shift. We also show how the C*-algebra of a twist may be identified with the Cuntz-Pimsner algebra constructed from a certain C*-correspondence.

On the Takai duality for $L^{p}$ operator crossed products

The aim of this paper is to study a problem raised by N. C. Phillips concerning the existence of Takai duality for $L^p$ operator crossed products $F^{p}(G,A,α)$, where $G$ is a locally compact Abelian group, $A$ is an $L^{p}$ operator algebra and $α$ is an isometric action of $G$ on $A$. Inspired by D. Williams' proof for the Takai duality theorem for crossed products of $C^*$-algebras, we construct a homomorphism $Φ$ from $F^{p}(\hat{G},F^p(G,A,α),\hatα)$ to $\mathcal{K}(l^{p}(G))\otimes_{p}A$ which is a natural $L^p$-analog of D. Williams' map. For countable discrete Abelian groups $G$ and …

Ext classes and embeddings for C*-algebras of graphs with sinks

We consider directed graphs E which have been obtained by adding a sink to a fixed graph G. We associate an element of Ext(C*(G)) to each such E, and show that the classes of two such graphs are equal in Ext(C*(G)) if and only if the associated C*-algebra of one can be embedded as a full corner in the C*-algebra of the other in a particular way. If every loop in G has an exit, then we are able to use this result to generalize some of the classification theorems of Raeburn, Tomforde, and Williams for C*-algebras of graphs with sinks.

Williams猜想对流星图成立

Williams' Conjecture holds for meteor graphs

本文证明流星图的移位等价、强移位等价、Leavitt路径代数的分次Morita等价及分次K-理论模同构彼此等价，并由此建立Leavitt路径代数与图C*-代数在Morita等价意义下的对应关系。

Williams猜想对Gelfand-Kirillov维数为3的图成立

Williams' conjecture holds for graphs of Gelfand-Kirillov dimension three

该研究针对Gelfand-Kirillov维数为3的图，给出其强移位等价的数论准则，证明移位等价与强移位等价、Leavitt路代数分次Morita等价、分次K理论保序模同构等价，并得出相关代数与范畴等价的推论，验证Williams猜想成立。

The primitive ideal space of the partial-isometric crossed product of a system by a single automorphism

Let $(A,α)$ be a system consisting of a $C^*$-algebra $A$ and an automorphism $α$ of $A$. We describe the primitive ideal space of the partial-isometric crossed product $A\times_α^{\textrm{piso}}\mathbb{N}$ of the system by using its realization as a full corner of a classical crossed product and applying some results of Williams and Echterhoff.

具有阿贝尔迷向群的Étale群胚C*-代数的本原谱

The primitive spectrum of C*-algebras of etale groupoids with abelian isotropy

本文描述了Hausdorff局部紧étale群胚C*-代数本原谱的部分拓扑结构，在顺从且具有阿贝尔迷向群的条件下证明了van Wyk和Williams的猜想，完全确定了一大类可分C*-代数的理想结构。

Exhausting families of representations and spectra of pseudodifferential operators

Families of representations of suitable Banach algebras provide a powerful tool in the study of the spectral theory of (pseudo)differential operators and of their Fredholmness. We introduce the new concept of an exhausting family of representations of a C*-algebra A. An {\em exhausting family} of representations of a C*-algebra A is a set F of representations of A with the property that every irreducible representation of A is weakly contained in some ϕ\in F. An exhausting family F of representations of A has the property that `"a \in A is invertible if, and if, ϕ(a) is invertible for any ϕ\in…

通过 Cuntz-Krieger 代数的视角看移位等价

Shift equivalences through the lens of Cuntz-Krieger algebras

受 Williams 问题启发，本文通过研究图 C*-代数引入三种等价关系，在仅假设 SE 的条件下提供阻碍 SSE 的新方法，并据此测量 SSE 与 RE 的差异以反驳前人证明。

A remark on matrix product operator algebras, anyons and subfactors

We show that the mathematical structures in a recent work of Bultinck-Mariena-Williamson-Sahinoglu-Haegemana-Verstraete are the same as those of flat symmetric bi-unitary connections and the tube algebra in subfactor theory. More specifically, a system of flat symmetric bi-unitary connections arising from a subfactor with finite index and finite depth satisfies all their requirements for tensors and the tube algebra for such a subfactor and the anyon algebra for such tensors are isomorphic up to the normalization constants. Furthermore, the matrix product operator algebras arising from tensors…

Følner sequences in operator theory and operator algebras

The present article is a review of recent developments concerning the notion of Følner sequences both in operator theory and operator algebras. We also give a new direct proof that any essentially normal operator has an increasing Følner sequence $\{P_n\}$ of non-zero finite rank projections that strongly converges to 1. The proof is based on Brown-Douglas-Fillmore theory. We use Følner sequences to analyze the class of finite operators introduced by Williams in 1970. In the second part of this article we examine a procedure of approximating any amenable trace on a unital and separable C*-alge…

Wieler 螺线管：非 Hausdorff 扩张性、Cuntz–Pimsner 模型与函子性质

Wieler solenoids: non-Hausdorff expansiveness, Cuntz-Pimsner models, and functorial properties

本文研究由 Wieler 平稳逆极限诱导的 Fell 代数谱自映射，证明其构成紧致局部 Hausdorff 空间上的扩张满射局部同胚，并据此计算 Smale 空间稳定与不稳定 Ruelle 代数的 K-理论不变量。