Are Austad, Eduard Ortega
For a second-countable locally compact Hausdorff étale groupoid $\mathcal{G}$ with a continuous $2$-cocycle $σ$ we find conditions that guarantee that $\ell^1 (\mathcal{G},σ)$ has a unique $C^*$-norm.
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A going-down principle for {é}tale groupoids and the Baum-Connes conjecture
本文扩展了étale群胚的下行原理，通过群胚单纯复形研究证明了一般étale群胚的Baum-Connes装配映射分裂单射，并应用于拓扑K-理论连续性和Künneth公式。
Roe algebras and coarse index maps for spaces with proper actions of {é}tale groupoids. I
本文为étale群胚真作用空间建立等变Roe代数与K-理论框架，构造粗指标映射的接收函子，并证明满足条件的群胚单纯复形存在通用模。
KMS states on the $\mathrm{C}^*$-algebras of Fell bundles over {é}tale groupoids
本文证明了étale群胚上饱和Fell丛的全C*-代数上KMS态与迷向群上Fell丛C*-代数的可测态场之间的一一对应，建立了积分-分解定理，并通过群胚交叉积、矩阵代数等例子加以验证。
We investigate recent uniqueness theorems for reduced $C^*$-algebras of Hausdorff étale groupoids in the context of inverse semigroups. In many cases the distinguished subalgebra is closely related to the structure of the inverse semigroup. In order to apply our results to full $C^*$-algebras, we also investigate amenability. More specifically, we obtain conditions that guarantee amenability of the universal groupoid for certain classes of inverse semigroups. These conditions also imply the existence of a conditional expectation onto a canonical subalgebra.
We present a uniqueness theorem for the reduced C*-algebra of a twist $\mathcal{E}$ over a Hausdorff étale groupoid $\mathcal{G}$. We show that the interior $\mathcal{I}^\mathcal{E}$ of the isotropy of $\mathcal{E}$ is a twist over the interior $\mathcal{I}^\mathcal{G}$ of the isotropy of $\mathcal{G}$, and that the reduced twisted groupoid C*-algebra $C_r^*(\mathcal{I}^\mathcal{G}; \mathcal{I}^\mathcal{E})$ embeds in $C_r^*(\mathcal{G}; \mathcal{E})$. We also investigate the full and reduced twisted C*-algebras of the isotropy groups of $\mathcal{G}$, and we provide a sufficient condition und…
Given an ample, Hausdorff groupoid $\mathcal{G}$, and a unital commutative ring $R$, we consider the Steinberg algebra $A_R(\mathcal {G})$. First we prove a uniqueness theorem for this algebra and then, when $\mathcal{G}$ is graded by a cocycle, we study graded ideals in $A_R(\mathcal {G})$. Applications are given for two classes of ample groupoids, namely those coming from actions of groups on graphs, and also to groupoids defined in terms of Boolean dynamical systems.
In this paper, we introduce quotients of étale groupoids. Using the notion of quotients, we describe the abelianizations of groupoid C*-algebras. As another application, we obtain a simple proof that effectiveness of an étale groupoid is implied by the full uniqueness property of its groupoid C*-algebra.
An ultragraph gives rise to a labelled graph with some particular properties. In this paper we describe the algebras associated to such labelled graphs as groupoid algebras. More precisely, we show that the known groupoid algebra realization of ultragraph C*-algebras is only valid for ultragraphs for which the range of each edge is finite, and we extend this realization to any ultragraph (including ultragraphs with sinks). Using our machinery, we characterize the shift space associated to an ultragraph as the tight spectrum of the inverse semigroup associated to the ultragraph (viewed as a lab…
We prove a uniqueness theorem and give a characterization of simplicity for Steinberg algebras associated to non-Hausdorff ample groupoids. We also prove a uniqueness theorem and give a characterization of simplicity for the C*-algebra associated to non-Hausdorff étale groupoids. Then we show how our results apply in the setting of tight representations of inverse semigroups, groups acting on graphs, and self-similar actions. In particular, we show that C*-algebra and the complex Steinberg algebra of the self-similar action of the Grigorchuk group are simple but the Steinberg algebra with coef…
We consider the boundary-path groupoids of topological higher-rank graphs. We show that the all such groupoids are topologically amenable. We deduce that the C*-algebras of topological higher-rank graphs are nuclear and prove versions of the gauge-invariant uniqueness theorem and the Cuntz-Krieger uniqueness theorem. We then provide a necessary and sufficient condition for simplicity of a topological higher-rank graph C*-algebra, and a condition under which it is also purely infinite.
We consider groupoids constructed from a finite number of commuting local homeomorphisms acting on a compact metric space, and study generalized Ruelle operators and $ C^{\ast} $-algebras associated to these groupoids. We provide a new characterization of $ 1 $-cocycles on these groupoids taking values in a locally compact abelian group, given in terms of $ k $-tuples of continuous functions on the unit space satisfying certain canonical identities. Using this, we develop an extended Ruelle-Perron-Frobenius theory for dynamical systems of several commuting operators ($ k $-Ruelle triples and c…
The groupoid approach to equilibrium states on right LCM semigroup C*-algebras
本文用群胚方法研究右LCM半群C*-代数上由同态N诱导的KMSβ-态，给出存在唯一性的充要条件，并在ζ函数有限时建立与核子代数迹态的显式双射。
Let G be a locally compact, Hausdorff groupoid in which s is a local homeomorphism and the unit space is totally disconnected. Assume there is a continuous cocycle c from G into a discrete group $Γ$. We show that the collection A(G) of locally-constant, compactly supported functions on G is a dense *-subalgebra of C_c(G) and that it is universal for algebraic representations of the collection of compact open bisections of G. We also show that if G is the groupoid associated to a row-finite graph or k-graph with no sources, then A(G) is isomorphic to the associated Leavitt path algebra or Kumji…
The ideal intersection property for essential groupoid C*-algebras
本文刻画了拓扑传递étale群胚的本质群胚C*-代数具有理想交性质的若干等价条件，包括不存在本质上受限的可服从各向同性截面和等变伪期望唯一性，并引入群胚在C*-代数上作用的新概念，给出Thompson群拟正则表示在Cuntz代数中的相对Powers平均性质应用。
We introduce twisted Steinberg algebras over a commutative unital ring $R$. These generalise Steinberg algebras and are a purely algebraic analogue of Renault's twisted groupoid C*-algebras. In particular, for each ample Hausdorff groupoid $G$ and each locally constant $2$-cocycle $σ$ on $G$ taking values in the units $R^\times$, we study the algebra $A_R(G,σ)$ consisting of locally constant compactly supported $R$-valued functions on $G$, with convolution and involution "twisted" by $σ$. We also introduce a "discretised" analogue of a twist $Σ$ over a Hausdorff étale groupoid $G$, and we show…
The Full Set of KMS-States for Abelian Kitaev Models
本文研究了阿贝尔基塔耶夫模型的KMS态，通过证明生成子代数与UHF代数的包含关系，构建了群oid C*代数，并证明了在β≥0时KMS态的唯一性及β→∞时极限态与无 frustrations 地面态一致。
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