Thomas Timmermann
We propose a definition of compact quantum groupoids in the setting of C*-algebras, associate to such a quantum groupoid a regular C*-pseudo-multiplicative unitary, and use this unitary to construct a dual Hopf C*-bimodule and to pass to a measurable quantum groupoid in the sense of Enock and Lesieur. Moreover, we discuss examples related to compact and to étale groupoids and study principal compact C*-quantum groupoids.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
The theory of measured quantum groupoids, as defined by Lesieur and myself, was made to generalize the theory of quantum groups made by Kustarmans and Vaes, but was only defined in a von Neumann algebra setting; Th. Timmermann constructed locally compact quantum groupoids, which is a C*-version of quantum groupoids. Here, we associate to such a locally compact quantum groupoid a measured quantum groupoid in which it is weakly dense; we then associate to a measured quantum groupoid a locally compact quantum groupoid which is weakly dense in the measured quantum groupoid, but such a locally comp…
In a series of papers, we have shown that from the \repn theory of a compact groupoid one can reconstruct the groupoid using the procedure similar to the Tannaka-Krein duality for compact groups. In this part we introduce the Tannaka groupoid of a compact groupoid and show how to recover the original groupoid from its Tannaka groupoid.
Algebraic quantum transformation groupoids of compact type
本文引入了紧型代数量子变换广群及其离散型对偶，恢复了紧与离散Van Daele代数量子群间的对偶，并提供了源于Fell丛和量子商空间的实例。
In a series of papers, we have shown that from the representation theory of a compact groupoid one can reconstruct the groupoid using the procedure similar to the Tannaka-Krein duality for compact groups. In this part we study the Fourier and Fourier-Plancherel transforms and prove the Plancherel theorem for compact groupoids. We also study the central functions in the algebra of square integrable functions on the isotropy groups.
Generalizing the notion of matched pair of groups, we define and study matched pairs of locally compact groupoids endowed with Haar systems, in order to give new examples of measured quantum groupoids.
In a series of papers, we have shown that from the representatio theory of a compact groupoid one can reconstruct the groupoid using the procedure similar to the Tannaka-Krein duality for compact groups. In this part we study continuous representations of compact groupoids. We show that irreducible representations have finite dimensional fibres. We prove the Schur's lemma, Gelfand-Raikov theorem and Peter-Weyl theorem for compact groupoids.
A compactification of Fell is applied to locally compact non-Hausdorff groupoids and yields locally compact Hausdorff groupoids. In the etale case, this construction provides a geometric picture for the left-regular representations introduced by Khoshkam and Skandalis.
Dana P. Williams raised in [Proc. Am. Math. Soc., Ser. B, 2016] the following question: Must a second countable, locally compact, transitive groupoid have open range map? This paper gives a negative answer to that question. Although a second countable, locally compact transitive groupoid G may fail to have open range map, we prove that we can replace its topology with a topology which is also second countable, locally compact, and with respect to which G is a topological groupoid whose range map is open. Moreover, the two topologies generate the same Borel structure and coincide on the fibres …
We show how the C*-algebras of quantum complex projective spaces (standard or nonstandard) are related to groupoids.
In this article, when G is a locally compact quantum group, we associate to a braided-commutative G-Yetter-Drinfel'd algebra $(N,a,\hat{a})$ equipped with a normal faithful semi-finite weight verifying some appropriate condition, a structure of a measured quantum groupoid. The dual structure is then given by $(N,\hat{a},a)$. Examples are given, especially the situation of a quotient type co-ideal of a compact quantum group. This construction generalizes the standard construction of a transformation groupoid. Most of the results were announced by the second author in 2011, at a conference in Wa…
In this paper, we introduce properties including groupoid comparison, pure infiniteness and paradoxical comparison as well as a new algebraic tool called groupoid semigroup for locally compact Hausdorff étale groupoids. We show these new tools help establishing pure infiniteness of reduced groupoid $C^*$-algebras. As an application, we show a dichotomy of stably finiteness against pure infiniteness for reduced groupoid $C^*$-algebras arising from locally compact Hausdorff étale minimal topological principal groupoids. This generalizes the dichotomy obtained by Bönicke-Li and Rainone-Sims. We a…
We study coactions of finite quantum groupoids on unital $C^*$-algebras and obtain the Tannaka-Krein reconstruction theorem for them.
We show that the C*-algebra of a quantum sphere $C(S_{q}^{2n+1})$ can be realized as a groupoid C*-algebra of a groupoid which is explicitly identified and is independent of the parameter $q$.
Lie groupoids, the Satake compactification and the tempered dual, I: The Satake groupoid
本文提出Satake群oid的三种不同视角，通过拓扑、李理论和几何方法，为Harish-Chandra原理提供新证明。
We introduce and investigate using Hilbert modules the properties of the Fourier algebra A(G) for a locally compact groupoid G. We establish a duality theorem for such groupoids in terms of multiplicative module maps. This includes as a special case the classical duality theorem for locally compact groups proved by P. Eymard.
The purpose of this paper is extend the notion of morphism of groupoids introduced by Zakrzewski to locally compact $σ$-compact groupoids endowed with Haar systems and to use the extension to construct a covariant functor from this category to a category of $C^{\ast}$-algebras in the sense of Woronowicz.
The crystal groupoid of a compact semisimple Lie group and its flag varieties
本文引入紧半单李群的晶体群胚，证明其代数和轨道空间分别实现晶体极限和Weyl群，并推广至旗流形。
In this work we give a generalization of matched pairs of (finite) groups to describe a general class of depth two inclusions of factor von Neumann algebras and the C*-quantum groupoids associated with, using double groupoids.
We introduce and investigate using Hilbert modules the properties of the {\em Fourier algebra} $A(G)$ for a locally compact groupoid $G$. We establish a duality theorem for such groupoids in terms of multiplicative module maps. This includes as a special case the classical duality theorem for locally compact groups proved by P. Eymard.
We give a survey of the theory of finite quantum groupoids (weak Hopf algebras), including foundations of the theory and applications to finite depth subfactors, dynamical deformations of quantum groups, and invariants of knots and 3-manifolds.
We show that Haefliger's cohomology for etale groupoids, Moore's cohomology for locally compact groups and the Brauer group of a locally compact groupoid are all particular cases of sheaf (or Cech) cohomology for topological simplicial spaces.
We establish a Galois correspondence for finite quantum groupoid actions on II_1 factors and show that every finite index and finite depth subfactor is an intermediate subalgebra of a quantum groupoid crossed product. Moreover, any such a subfactor is completely and canonically determined by a quantum groupoid and its coideal *-subalgebra. This allows to express the bimodule category of a subfactor in terms of the representation category of a corresponding quantum groupoid and the principal graph as the Bratteli diagram of an inclusion of certain C^*-algebras related to it.
This is Part II in our multi-part series of papers developing the theory of a subclass of locally compact quantum groupoids ("quantum groupoids of separable type"), based on the purely algebraic notion of weak multiplier Hopf algebras. The definition was given in Part I. The existence of a certain canonical idempotent element $E$ plays a central role. In this Part II, we develop the main theory, discussing the structure of our quantum groupoids. We will construct from the defining axioms the right/left regular representations and the antipode map.
We define the Furstenberg boundary of a locally compact Hausdorff étale groupoid, generalising the Furstenberg boundary for discrete groups, by providing a construction of a groupoid-equivariant injective envelope. Using this injective envelope, we establish the absence of recurrent amenable subgroups in the isotropy as a sufficient criterion for the intersection property of a locally compact Hausdorff étale groupoid with compact unit space and no fixed points. This yields a criterion for C*-simplicity of minimal groupoids.
We give a definition of amenability at infinity for a locally compact, $σ$-compact and Hausdorff etale groupoid and we study in some case the exactness of the reduced $C^*$-algebra of a such groupoid.
In this series of papers, we develop the theory of a class of locally compact quantum groupoids, which is motivated by the purely algebraic notion of weak multiplier Hopf algebras. In this Part I, we provide motivation and formulate the definition in the C*-algebra framework. Existence of a certain canonical idempotent element is required and it plays a fundamental role, including the establishment of the coassociativity of the comultiplication. This class contains locally compact quantum groups as a subclass.
It is proved that every second countable locally Hausdorff and locally compact continuous groupoid has a Borel set of units that meets every orbit and is what is called "lacunary," a property that implies that the intersection with every orbit is countable.
We establish the equivalence of three versions of a finite dimensional quantum groupoid: a generalized Kac algebra introduced by T. Yamanouchi, a weak $C^*$-Hopf algebra introduced by G. Bohm, F. Nill and K. Szlachanyi (with an involutive antipode), and a Kac bimodule -- an algebraic version of a Hopf bimodule, the notion introduced by J.-M. Vallin. We also study the structure and construct examples of finite dimensional quantum groupoids.
In a recent article of Kenny De Commer, was investigated a Morita equivalence between locally compact quantum groups, in which a measured quantum groupoid, of basis $\mathbb{C}^2$, was constructed as a linking object. Here, we generalize all these constructions and concepts to the level of measured quantum groupoids. As for locally compact quantum groups, we apply this construction to the deformation of a measured quantum groupoid by a 2-cocycle.
Mimicking the von Neumann version of Kustermans and Vaes' locally compact quantum groups, Franck Lesieur had introduced a notion of measured quantum groupoid, in the setting of von Neumann algebras. In this article, we suppose that the basis of the measured quantum groupoid is central; in that case, we prove that a specific sub-${\bf C}^*$ algebra is invariant under all the data of the measured quantum groupoid; moreover, this sub-${\bf C}^*$-algebra is a continuous field of ${\bf C}^*$-algebras; when the basis is central in both the measured quantum groupoid and its dual, we get that the meas…
The Fourier-Stieltjes and Fourier algebras B(G), A(G) for a general locally compact group G, first studied by P. Eymard, have played an important role in harmonic analysis and in the study of operator algebras generated by G. Recently, there has been interest in developing versions of these algebras for locally compact groupoids, justification being that, just as in the group case, the algebras should play a useful role in the study of groupoid operator algebras. Versions of these algebras for the locally compact groupoid case appear in three related theories: (1) a measured groupoid theory (J…
Let N_0 \subset N_1 a depth 2, finite index inclusion of type II1 factors and N_0 \subset N_1 \subset N_2 \subset N_3 ... the corresponding Jones tower. D. Nikshych et L. Vainerman built dual structures of quantum C*-groupoid on the relative commutants N'_0 \cap N_2 et N'_1 \cap N_3. Here I define a new duality which allows a symetric construction without changing the involution. So the Temperley-Lieb algebras are selfdual quantum C*-groupoids and the quantum C*-groupoids associated to a finite depth finite index inclusion can be choosen selfdual. I show that every finite-dimensional connexe q…
This paper studies injective envelopes of groupoid dynamical systems and the corresponding boundaries. Analogue to the group case, we associate a bundle of compact Hausdorff spaces to any (discrete) groupoid (the Hamana boundary of the groupoid). We study bundles of compact topological spaces equipped with an action of a groupoid. We show that any groupoid has a minimal boundary (the Furstenberg boundary of the groupoid). We prove that the Hamana and Furstenberg boundaries are the same, for (discrete) groupoids. We find the relation between the reduced crossed product of the $\G$-injective env…
For second countable locally compact Hausdorff groupoids, the property of possessing a Haar system is preserved by equivalence.
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。
Quantum metrics from length functions on étale groupoids
本文探讨如何通过 proper 连续长度函数构造紧量子度量空间，并提出利用 compactly 支持的傅里叶乘子作为验证条件，为理解 unital AF 群代数的量子度量几何提供了群oid 方法。
We extend the symbol calculus and study the limit operator theory for $σ$-compact, étale and amenable groupoids, in the Hilbert space case. This approach not only unifies various existing results which include the cases of exact groups and discrete metric spaces with Property A, but also establish new limit operator theories for group/groupoid actions and uniform Roe algebras of groupoids. In the process, we extend a monumental result by Exel, Nistor and Prudhon, showing that the invertibility of an element in the groupoid $C^*$-algebra of a $σ$-compact amenable groupoid with a Haar system is …
This article extends the main results of the publication arXiv:2001.01312 to the case of a twisted groupoid. More precisely, it gives a decomposition of the C*-algebra of a twisted locally compact groupoid with Haar system in presence of a normal subgroupoid. When the normal subgroupoid and the twist over it are abelian, one obtains another twisted groupoid C*-algebra. This is applied to C*-algebraic deformation quantization and to multiplier representations of locally compact abelian groups.
We provide an exposition and proof of Renault's equivalence theorem for crossed products by locally Hausdorff, locally compact groupoids. Our approach stresses the bundle approach, concrete imprimitivity bimodules and is a preamble to a detailed treatment of the Brauer semigroup for a locally Hausdorff, locally compact groupoid.
Franck Lesieur had introduced in his thesis (now published in an expended and revised version in the {\it Mémoires de la SMF} (2007)) a notion of measured quantum groupoid, in the setting of von Neumann algebras and a simplification of Lesieur's axioms is presented in an appendix of this article. We here develop the notions of actions, crossed-product, and obtain a biduality theorem, following what had been done by Stefaan Vaes for locally compact quantum groups. Moreover, we prove that the inclusion of the initial algebra into its crossed-product is depth 2, which gives a converse of a result…
This paper gives a first step toward extending the theory of Fourier-Stieltjes algebras from groups to groupoids. If G is a locally compact (second countable) groupoid, we show that B(G), the linear span of the Borel positive definite functions on G, is a Banach algebra when represented as an algebra of completely bounded maps on a C^*-algebra associated with G. This necessarily involves identifying equivalent elements of B(G). An example shows that the linear span of the continuous positive definite functions need not be complete. For groups, B(G) is isometric to the Banach space dual of C^*(…
Let G be a (not necessarily Hausdorff) locally compact groupoid. We introduce a notion of properness for G, which is invariant under Morita-equivalence. We show that any generalized morphism between two locally compact groupoids which satisfies some properness conditions induces a C*-correspondence from $C*_r(G_1)$ to $C^*_r(G_2)$, and thus two Morita equivalent groupoids have Morita-equivalent $C^*$-algebras.
We construct a locally compact groupoid with the properties in the title. Our example is based closely on constructions used by Higson, Lafforgue, and Skandalis in their work on counterexamples to the Baum-Connes conjecture. It is a bundle of countable groups over the one point compactification of the natural numbers, and is Hausdorff, second countable and étale.
Suppose G is a second countable, locally compact, Hausdorff, principal groupoid with a fixed left Haar system. We define a notion of integrability for groupoids, and show G is integrable if and only if the groupoid C*-algebra C*(G) has bounded trace.
It is a well-known result of Eymard that the Fourier-Stieltjes algebra of a locally compact group $G$ can be identified with the dual of the group $\cs$ $C^{*}(G)$. A corresponding result for a locally compact groupoid $G$ has been investigated by Renault, Ramsay and Walter. We show that the Fourier-Stieltjes algebra $B_μ(G)$ of $G$ (with respect to a quasi-invariant measure $μ$ on the unit space $X$ of $G$) can be characterized in operator space terms as the dual of the Haagerup tensor product $\ov{L^{2}(X,μ)}^{r}\otimes_{hA}C^{*}(G,μ)\otimes_{h A}L^2(X,μ)^c$ and as the space of completely bo…
It is shown that a locally compact groupoid with open range map does not always admit a Haar system. It then is shown how to construct a Haar system if the stability groupoid and the quotient by the stability groupoid both admit one.
Groupoid exactness and the weak containment problem
研究局部紧群体中精确性的类比，证明六个精确性概念的等价性，并探讨群体可接受性与C*-代数关系。
Mimicking the von Neumann version of Kustermans and Vaes' locally compact quantum groups, Franck Lesieur had introduced a notion of measured quantum groupoid, in the setting of von Neumann algebras. In a former article, the author had introduced the notion of actions, crossed-product, dual actions of a measured quantum groupoid: a biduality theorem for actions had been proved. This article continues that program : we prove the existence of a standard implementation for an action, and a bidulaity theorem for weights. We generalize this way results which were proved, for locally compact quantum …
Generalised Twisted Groupoids and their C*-algebras
该研究考虑局部紧豪斯多夫群胚$G$，通过更一般的局部紧豪斯多夫阿贝尔群$Γ$而非复单位圆$\mathbb{T}$进行扭曲，仿照通常扭曲群胚C*-代数构造其C*-代数，并证明任何$Γ$-扭曲群胚C*-代数同构于通常的扭曲群胚C*-代数。
Amenable unitary representations of locally compact groupoids
本文研究局部紧群体的可约酉表示，通过Hilbert束理论和算子值向量测度定义拓扑不变均值，将Bekka对群的可约性刻画推广到群体。
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…
Let $G$ be second countable locally compact Hausdorff groupoid with a continuous Haar system. We remove the assumption of amenability in a theorem by Clark about GCR groupoid $C^*$-algebras. We show that if the groupoid $C^*$-algebra of $G$ is GCR then the orbits of $G$ are locally closed.
Let G be second countable locally compact Hausdorff groupoid with a continuous Haar system. We remove the assumption of amenability in a theorem of Clark about groupoids whose $C^*$-algebras are CCR. We show that if the groupoid C*-algebra of G is CCR then the orbits of G are closed.
Suppose $\mathcal{G}$ is a second-countable locally compact Hausdorff étale groupoid, $G$ is a discrete group containing a unital subsemigroup $P$, and $c:\mathcal{G}\rightarrow G$ is a continuous cocycle. We derive conditions on the cocycle such that the reduced groupoid $C^*$-algebra $C_r^*(\mathcal{G})$ may be realised naturally as the covariance algebra of a product system over $P$ with coefficient algebra $C_r^*(c^{-1}(e))$. When $(G,P)$ is a quasi-lattice ordered group, we also derive conditions that allow $C_r^*(\mathcal{G})$ to be realised as the Cuntz--Nica--Pimsner algebra of a compa…
Co-universal $C^{\ast}$-algebras for product systems over finite aligned subcategories of groupoids
本文研究了左可消小范畴上的乘积系统，引入了紧对齐乘积系统及其Nica协变表示的概念，并证明了群胚有限对齐子范畴上该系统余泛代数的存在性。
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.
Consider the Deaconu-Renault groupoid of an action of a finitely generated free abelian monoid by local homeomorphisms of a locally compact Hausdorff space. We catalogue the primitive ideals of the associated groupoid C*-algebra. For a special class of actions we describe the Jacobson topology.
In this paper, we consider topological semigroup actions on compact topological spaces. Under mild assumptions on the semigroup and the action, we construct a semi-direct product groupoid with a Haar system. We also show that it is equivalent to a transformation groupoid. We apply this construction to the Wiener-Hopf $C^{*}$-algebras.
Mimicking a recent article of Stefaan Vaes, in which was proved that every locally compact quantum group can act outerly, we prove that we get the same result for measured quantum groupoids, with an appropriate definition of outer actions of measured quantum groupoids. This result is used to show that every measured quantum groupoid can be found from some depth 2 inclusion of von Neumann algebras.
The stabilization theorem for $A$-Hilbert modules was established by G. G. Kasparov. The equivariant version, in which a locally compact group $H$ acts properly on a locally compact space $Y$, was proved by N. C. Phillips. This equivariant theorem involves the Hilbert $(H,C_{0}(Y))$-module $C_{0}(Y,L^{2}(H)^{\infty})$. It can naturally be interpreted in terms of a stabilization theorem for proper groupoids, and the paper establishes this theorem within the general proper groupoid context. The theorem has applications in equivariant KK-theory and groupoid index theory.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。