Ruy Exel
We show that a (not necessarily Hausdorff) etale, second countable groupoid G with totally disconnected unit space may be reconstructed solely from the algebraic structure of its ample semigroup S. We also show that C*(G) possesses a universal property related to tight representations of S.
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Ample groupoids that are neither almost finite nor purely infinite
研究最小富足变换群状体既非近有限亦非纯无限的问题，构造了非近有限且非纯无限的富足群状体，不源于变换群状体。
Groupoid strict comparison for Ample Groupoids: Fiberwise Supramenability and Topological Amenability
本文为局部紧Hausdorff étale群胚引入纤维上超可支配性，证明在最小充足群胚中，若满足纤维上超可支配性或拓扑可amenable性，则其闭开型半群几乎无穿孔，进而σ-紧时具有群胚严格比较，并应用于Matui的AH猜想及纯无穷性。
We study the topological full group of ample groupoids over locally compact spaces. We extend Matui's definition of the topological full group from the compact, to the locally compact case. We provide two general classes of groupoids for which the topological full group, as an abstract group, is a complete isomorphism invariant. Hereby extending Matui's Isomorphism Theorem. As an application, we study graph groupoids and their topological full groups, and obtain sharper results for this class. The machinery developed in this process is used to prove an embedding theorem for ample groupoids, ak…
Cup and cap products for cohomology and homology groups of ample groupoids
本文研究丰富群胚上同调与同调中的杯积和帽积，给出清晰定义，并应用帽积分析群胚C*-代数自同构的渐近内性，首次显式计算平铺空间上同调中的杯积。
The Almost Finite--Purely Infinite Dichotomy for Minimal Amenable Ample Groupoids
本文证明极小主拓扑顺从 ample 群胚在存在不变概率测度时强几乎有限，否则纯无限，并证明有界度均匀 Borel 顺从 Følner 图的 Borel 几乎有限性。
For almost finite groupoids, we study how their homology groups reflect dynamical properties of their topological full groups. It is shown that two clopen subsets of the unit space has the same class in H_0 if and only if there exists an element in the topological full group which maps one to the other. It is also shown that a natural homomorphism, called the index map, from the topological full group to H_1 is surjective and any element of the kernel can be written as a product of four elements of finite order. In particular, the index map induces a homomorphism from H_1 to K_1 of the groupoi…
We prove that a minimal second countable ample groupoid has dynamical comparison if and only if its type semigroup is almost unperforated. Moreover, we investigate to what extent a not necessarily minimal almost finite groupoid has an almost unperforated type semigroup. Finally, we build a bridge between coarse geometry and topological dynamics by characterizing almost finiteness of the coarse groupoid in terms of a new coarsely invariant property for metric spaces, which might be of independent interest in coarse geometry. As a consequence, we are able to construct new examples of almost fini…
Given an ample groupoid, we construct a spectral sequence with groupoid homology with integer coefficients on the second sheet, converging to the K-groups of the (reduced) groupoid C*-algebra, provided the groupoid has torsion-free stabilizers and satisfies a strong form of the Baum-Connes conjecture. The construction is based on the triangulated category approach to the Baum-Connes conjecture developed by Meyer and Nest. We also present a few applications to topological dynamics and discuss the HK conjecture of Matui.
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.
Ample groupoid homology and étale correspondences
本文证明充裕群胚间的平展对应诱导同调群同态，并通过子群胚的归纳—限制伴随构造任意群胚模的消解，以 Tor 图景显式刻画相应同调映射。
Let $\mathcal{G}$ be a locally compact $σ$-compact Hausdorff ample groupoid on a compact space. In this paper, we further examine the (ubiquitous) fiberwise amenability introduced by the author and Jianchao Wu for $\mathcal{G}$. We define the corresponding concepts of Følner sequences and Banach densities for $\mathcal{G}$, based on which, we establish a topological groupoid version of the Ornstein-Weiss quasi-tilling theorem. This leads to the notion of almost finiteness in measure for ample groupoids as a weaker version of Matui's almost finiteness. As applications, we first show that $C^*_r…
In this paper, we study the ideal structure of reduced $C^*$-algebras $C^*_r(G)$ associated to étale groupoids $G$. In particular, we characterize when there is a one-to-one correspondence between the closed, two-sided ideals in $C_r^*(G)$ and the open invariant subsets of the unit space $G^{(0)}$ of $G$. As a consequence, we show that if $G$ is an inner exact, essentially principal, ample groupoid, then $C_r^*(G)$ is (strongly) purely infinite if and only if every non-zero projection in $C_0(G^{(0)})$ is properly infinite in $C_r^*(G)$. We also establish a sufficient condition on the ample gr…
We study a Going-Down (or restriction) principle for ample groupoids and its applications. The Going-Down principle for locally compact groups was developed by Chabert, Echterhoff and Oyono-Oyono and allows to study certain functors, that arise in the context of the topological K-theory of a locally compact group, in terms of their restrictions to compact subgroups. We extend this principle to the class of ample Hausdorff groupoids using Le Gall's groupoid equivariant version of Kasparov's bivariant KK-theory. Moreover, we provide an application to the Baum-Connes conjecture for ample groupoid…
A note on Boolean inverse monoids and ample groupoids
本文研究布尔逆幺半群与 ample 群胚之间的联系，通过具体构造和性质分析，揭示了二者之间的对应关系。
This paper investigates the $\mathrm{K}$-theory of twisted groupoid $\mathrm{C}^*$-algebras. It is shown that a homotopy of twists on an ample groupoid satisfying the Baum-Connes conjecture with coefficients gives rise to an isomorphism between the $\mathrm{K}$-theory groups of the respective twisted groupoid $\mathrm{C}^*$-algebras. The results are also interpreted in an inverse semigroup setting and applied to generalized Renault-Deaconu groupoids and $P$-graph algebras.
We investigate the homology of ample Hausdorff groupoids. We establish that a number of notions of equivalence of groupoids appearing in the literature coincide for ample Hausdorff groupoids, and deduce that they all preserve groupoid homology. We compute the homology of a Deaconu{Renault groupoid associated to k pairwisecommuting local homeomorphisms of a zero-dimensional space, and show that Matui's HK conjecture holds for such a groupoid when k is one or two. We specialise to k-graph groupoids, and show that their homology can be computed in terms of the adjacency matrices, using a chain co…
We give algebraic characterizations of the type I and CCR properties for locally compact second countable, ample Hausdorff groupoids in terms of subquotients of their Boolean inverse semigroups of compact open bisections. It yields in turn algebraic characterizations of both properties for inverse semigroups in terms of subquotients of their booleanizations.
Ample groupoids, topological full groups, algebraic K-theory spectra and infinite loop spaces
本文受Higman-Thompson群同调研究启发，通过构造紧开双截面的小置换范畴，建立了丰富群胚、拓扑全群、代数K-理论谱与无穷环空间间的联系，并应用于计算同调不变量、证明消没及无环性结果，最终证明了Matui的AH猜想。
Rigidity of pseudofunction algebras of ample groupoids
本文证明 Hausdorff 充足广群可由其伪函数及约化 $L^p$-算子代数完全恢复，并通过 Moore–Penrose 可逆部分等距构造逆半群，同时验证 Rakočević 猜想。
Discretisation and independent resolutions of ample groupoids
提出基于分立化和独立分解的框架，通过将充分广群等价于离散广群，并构建独立分解，系统研究其同调与K理论，应用于Garside范畴等例子。
Cohomology of ample groupoids
该研究为充裕广群引入基于平坦消解的上链复形，证明其与已有连续上闭上同调一致且在Morita等价下不变，推导了斜积上同调的正合序列，给出了AF-广群等的上同调计算方法。
Chern character for torsion-free ample groupoids
本文构建了从 Baum-Connes 组合映射域到 Crainic-Moerdijk 群oids 带有有理系数的同调群的 Chern 符号映射，验证了 Matui 的 HK 猜想变体，并应用于超几何动力系统、拓扑满群同调等。
A homological characterization of AF groupoids
本文通过同调维数零的特征，研究了AF群体的结构，证明了其与同调维数零的等价性，并推广到单位环上的群体。
Functoriality and Weyl Groupoids of Ample C*-Diagonal Pairs
本文研究丰富C*-对角对及其Weyl群胚的函子性，通过构造诱导的部分群胚态射，分析其性质，并应用于理想、条件期望和动态比较，证明Weyl函子在非扭曲对子范畴上的忠实性。
On von Neumann regularity of ample groupoid algebras
本文完全刻画了充足群胚代数及其分次版本的冯·诺依曼正则性，解决了逆半群代数正则性这一自1970年悬而未决的问题，并推广了Leavitt路代数等相关结果。
Stable finiteness of ample groupoid algebras, traces and applications
本文研究充足群胚代数的稳定有限性，应用于逆半群和Leavitt路径代数，发展迹理论，并证明Hausdorff充足群胚的稳定有限性蕴含约化C*-代数上有迹态。
Representation theory of topological full groups of étale groupoids and paradoxicality
研究拓扑满群的表示论，探讨其可计算性及轨道、Koopman和群oid左正则表示的比较，统一了关于半群和群oid悖论性的最新结果。
On groupoids beyond partial actions, inner amenability, and models for Kirchberg algebras
本文构造了首个非内可amenable且非部分作用变换群oid的例子，解决了Anantharaman--Delaroche和Exel的问题，并展示了Deaconu--Renault群oid不来源于离散群的部分作用，同时提供了变换群oid模型。
Given an action $φ$ of of inverse semigroup $S$ on a ring $A$ (with domain of $φ(s)$ denoted by $D_{s^*}$) we show that if the ideals $D_e$, with $e$ an idempotent, are unital, then the skew inverse semigroup ring $A\rtimes S$ can be realized as the convolution algebra of an ample groupoid with coefficients in a sheaf of rings. Conversely, we show that the convolution algebra of an ample groupoid with coefficients in a sheaf of rings is isomorphic to a skew inverse semigroup ring of this sort. We recover known results in the literature for Steinberg algebras over a field as special cases.
We study the finite versus infinite nature of C*-algebras arising from etale groupoids. For an ample groupoid G, we relate infiniteness of the reduced C*-algebra of G to notions of paradoxicality of a K-theoretic flavor. We construct a pre-ordered abelian monoid S(G) which generalizes the type semigroup introduced by Rørdam and Sierakowski for totally disconnected discrete transformation groups. This monoid reflects the finite/infinite nature of the reduced groupoid C*-algebra of G. If G is ample, minimal, and topologically principal, and S(G) is almost unperforated we obtain a dichotomy betwe…
Homology and K-theory of dynamical systems. II. Smale spaces with totally disconnected transversal
本文将群胚同调与 K-理论关系应用于 Smale 空间，证明具有全不交稳定集的 Smale 空间相关谱序列在第二页展示 Putnam 稳定同调群，且该同调同构于不稳定等价关系的群胚同调。
We define a bundle over a totally disconnected set such that each fiber is homeomorphic to a fractal blowup. We prove that there is a natural action of a Renault-Deaconu groupoid on our fractafold bundle and that the resulting action groupoid is a Renault-Deaconu groupoid itself. We also show that when the bundle is locally compact the associated $C^*$-algebra is primitive and has a densely defined lower-semicontinuous trace.
Twisted étale groupoid algebras have been studied recently in the algebraic setting by several authors in connection with an abstract theory of Cartan pairs of rings. In this paper, we show that extensions of ample groupoids correspond in a precise manner to extensions of Boolean inverse semigroups. In particular, discrete twists over ample groupoids correspond to certain abelian extensions of Boolean inverse semigroups and we show that they are classified by Lausch's second cohomology group of an inverse semigroup. The cohomology group structure corresponds to the Baer sum operation on twists…
Discrete Coefficients and Open Invariant Covers in the Homology of Ample Groupoids
本文建立了 ample 群体摩尔同调的广义系数定理和梅耶-维托里斯长正合列，通过链级证明提供了有效计算工具，强调系数群离散性的重要性。
We give an example of a locally compact effective Hausdorff, minimal ample groupoid such that its rational homology differs from the $K$-theory of its reduced groupoid $C^*$-algebra. Moreover, we prove that such example satisfies Matui's AH-conjecture.
Homology and K-theory of dynamical systems III. Beyond stably disconnected Smale spaces
本文通过引入空间方向分解，将Smale空间étale群胚的同调不变量研究推广到超越稳定不连通情形，证明Putnam同调与Crainic-Moerdijk群胚同调同构，并给出不稳定C*-代数K群的谱序列计算方法。
Partial actions of free groups and groupoid homology
研究自由群的偏作用群oid的同调计算及其代数的全局维数特性。
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.
We construct ample groupoids from certain categories of paths, and prove that their $C^*$-algebras coincide with the continued fraction AF algebras of Effros and Shen. The proof relies on recent classification results for simple nuclear $C^*$-algebras. The groupoids are not principal. This provides examples of Cartan subalgebras in the continued fraction AF algebras that are isomorphic, but not conjugate, to the standard diagonal subalgebras.
We show the reduced $C^*$-algebra of a graded ample groupoid is a strongly graded $C^*$-algebra if and only if the corresponding Steinberg algebra is a strongly graded ring. We apply this result to get a theorem about the Leavitt path algebra and $C^*$-algebra of an arbitrary graph.
Let $G$ be a Hausdorff, étale groupoid that is minimal and topologically principal. We show that $C^*_r(G)$ is purely infinite simple if and only if all the nonzero positive elements of $C_0(G^0)$ are infinite in $C_r^*(G)$. If $G$ is a Hausdorff, ample groupoid, then we show that $C^*_r(G)$ is purely infinite simple if and only if every nonzero projection in $C_0(G^0)$ is infinite in $C^*_r(G)$. We then show how this result applies to $k$-graph $C^*$-algebras. Finally, we investigate strongly purely infinite groupoid $C^*$-algebras.
We construct the full and reduced C*-algebras of an ample groupoid from its complex Steinberg algebra. We also show that our construction gives the same C*-algebras as the standard constructions. In the last section, we consider an arbitrary locally compact, second-countable, étale groupoid, possibly non-Hausdorff. Using the techniques developed for Steinberg algebras, we show that every $*$-homomorphism from Connes' space of functions to $B(\mathcal{H})$ is automatically I-norm bounded. Previously, this was only known for Hausdorff groupoids.
We unify various étale groupoid reconstruction theorems such as: 1) Kumjian-Renault's reconstruction from a groupoid C*-algebra. 2) Exel's reconstruction from an ample inverse semigroup. 3) Steinberg's reconstruction from a groupoid ring. 4) Choi-Gardella-Thiel's reconstruction from a groupoid L^p-algebra. We do this by working with certain bumpy semigroups S of functions defined on an étale groupoid G. The semigroup structure of S together with the diagonal subsemigroup D then yields a natural domination relation < on S. The groupoid of <-ultrafilters is then isomorphic to the original groupo…
In this paper we show that for an almost finite minimal ample groupoid $G$, its reduced $\mathrm{C}^*$-algebra $C_r^*(G)$ has real rank zero and strict comparison even though $C_r^*(G)$ may not be nuclear in general. Moreover, if we further assume $G$ being also second countable and non-elementary, then its Cuntz semigroup ${\rm Cu}(C_r^*(G))$ is almost divisible and ${\rm Cu}(C_r^*(G))$ and ${\rm Cu}(C_r^*(G)\otimes \mathcal{Z})$ are canonically order-isomorphic, where $\mathcal{Z}$ denotes the Jiang-Su algebra.
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 extend Matui's notion of almost finiteness to general etale groupoids and show that the reduced groupoid C*-algebras of minimal almost finite groupoids have stable rank one. The proof follows a new strategy, which can be regarded as a local version of the large subalgebra argument. The following three are the main consequences of our result. (i) For any group of (local) subexponential growth and for any its minimal action admitting a totally disconnected free factor, the crossed product has stable rank one. (ii) Any countable amenable group admits a minimal action on the Cantor set all whos…
We introduce a new variant of the coarse Baum-Connes conjecture designed to tackle coarsely disconnected metric spaces called the boundary coarse Baum-Connes conjecture. We prove this conjecture for many coarsely disconnected spaces that are known to be counterexamples to the coarse Baum-Connes conjecture. In particular, we give a geometric proof of this conjecture for spaces of graphs that have large girth and bounded vertex degree. We then connect the boundary conjecture to the coarse Baum-Connes conjecture using homological methods, which allows us to exhibit all the current uniformly discr…
We show how to reconstruct a graded ample Hausdorff groupoid with topologically principal neutrally graded component from the ring structure of its graded Steinberg algebra over any commutative integral domain with 1, together with the embedding of the canonical abelian subring of functions supported on the unit space. We deduce that diagonal-preserving ring isomorphism of Leavitt path algebras implies $C^*$-isomorphism of $C^*$-algebras for graphs $E$ and $F$ in which every cycle has an exit.
Chain Conditions for Etale Groupoid Algebras
本文针对艾达尔群胚代数，刻画了其范畴诺特/阿廷、局部诺特/阿廷及半单性质，推广了Leavitt路代数的相关链条件研究结果。
Given an ample groupoid $G$ with compact unit space, we study the canonical representation of the topological full group $[[G]]$ in the full groupoid $C^*$-algebra $C^*(G)$. In particular, we show that the image of this representation generates $C^*(G)$ if and only if $C^*(G)$ admits no tracial state. The techniques that we use include the notion of groups covering groupoids. As an application, we provide sufficient conditions for $C^*$-simplicity of certain topological full groups, including those associated with topologically free and minimal actions of non-amenable and countable groups on t…
The author has previously associated to each commutative ring with unit $\Bbbk$ and étale groupoid $\mathscr G$ with locally compact, Hausdorff, totally disconnected unit space a $\Bbbk$-algebra $\Bbbk\mathscr G$. The algebra $\Bbbk\mathscr G$ need not be unital, but it always has local units. The class of groupoid algebras includes group algebras, inverse semigroup algebras and Leavitt path algebras. In this paper we show that the category of unitary $\Bbbk\mathscr G$-modules is equivalent to the category of sheaves of $\Bbbk$-modules over $\mathscr G$. As a consequence we obtain a new proof …
We extend to arbitrary commutative base rings a recent result of Demeneghi that every ideal of an ample groupoid algebra over a field is an intersection of kernels of induced representations from isotropy groups, with a much shorter proof, by using the author's Disintegration Theorem for groupoid representations. We also prove that every primitive ideal is the kernel of an induced representation from an isotropy group; however, we are unable to show, in general, that it is the kernel of an irreducible induced representation. If each isotropy group is finite (e.g., if the groupoid is principal)…
In this paper, we prove that the algebra of an étale groupoid with totally disconnected unit space has a simple algebra over a field if and only if the groupoid is minimal and effective and the only function of the algebra that vanishes on every open subset is the null function. Previous work on the subject required the groupoid to be also topologically principal in the non-Hausdorff case, but we do not. Furthermore, we provide the first examples of minimal and effective but not topologically principal étale groupoids with totally disconnected unit spaces. Our examples come from self-similar g…
In this article we will study semigroupoids, and more specifically inverse semigroupoids. These are a common generalization to both inverse semigroups and groupoids, and provide a natural language on which several types of dynamical structures may be described. Moreover, this theory allows us to precisely compare and simultaneously generalize aspects of both the theories of inverse semigroups and groupoids. We begin by comparing and settling the differences between two notions of semigroupoids which appear in the literature (one by Tilson and another by Exel). We specialize this study to inver…
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…
Groupoid Homology and Classifying-Space Homology Are Not Isomorphic
本文通过康托尔集上的单位群胚，证明了Matui型群胚同调与分类空间奇异同调在0阶上不同构，前者为可数群而后者基数为2^ℵ₀。
Homology of Steinberg algebras
本文研究了ample群胚Steinberg代数的同调不变量，计算了特定群胚的Hochschild与循环同调，并利用Dennis迹建立了K理论到群胚同调的映射，最后针对Exel-Pardo群胚给出了具体计算。
Given an ample action of an inverse semigroup on a locally compact and Hausdorff topological space, we study the ideal structure of the crossed product algebra associated with it. By developing a theory of induced ideals, we manage to prove that every ideal in the crossed product algebra may be obtained as the intersection of ideals induced from isotropy group algebras. This can be interpreted as an algebraic version of the Effros-Hahn conjecture. Finally, as an application of our result, we study the ideal structure of a Steinberg algebra associated with an ample groupoid by interpreting it a…
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.
Banach algebras associated to twisted étale groupoids: inverse semigroup disintegration and representations on $L^p$-spaces
本文引入与扭曲étale群胚及扭曲逆半群作用相关的Banach代数，提供统一框架，证明分解定理，建立范数估计与层次结构，覆盖非Hausdorff情形，且结果对复C*代数有新贡献。
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