Madalina Roxana Buneci
We shall consider a locally compact groupoid endowed with a Haar system and having proper orbit space. We shall construct a groupoid C*-algebra which is independent of the Haar system (up to a *-isomorphism).
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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.
For second countable locally compact Hausdorff groupoids, the property of possessing a Haar system is preserved by equivalence.
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.
We consider groupoids on $\{1,2,..,d\}^\mathbb{N}$, cocycles and the counting measure as transverse function. We generalize results relating quasi-invariant probabilities with eigenprobabilities for the dual of the Ruelle operator. We assume a mild compatibility of the groupoid with the symbolic structure. We present a generalization of the Ruelle operator - the Haar-Ruelle operator - taking into account the Haar structure. We consider continuous and also Hölder cocycles. IFS with weights appears in our reasoning in the Hölder case.
Topological fundamental groupoid. III. Haar systems on the fundamental groupoid
本文研究道路连通、局部道路连通且半局部单连通空间的基本群胚上 Haar 系的存在性与描述，给出等价判据、C*代数描述及两个测度公式。
Induced Representation of Topological groupoids
介绍局部紧第二可数群胚\(G\)从闭宽子群胚\(H\)的连续酉表示的诱导表示，证明其基本性质、分步诱导定理等，还给出群胚版麦基张量积定理及紧可迁群胚版弗罗贝尼乌斯互反定理。
If $G$ is a second countable locally compact Hausdorff groupoid with Haar system, we show that every representation induced from an irreducible representation of a stability group is irreducible.
We define what it means for a proper continuous morphism between groupoids to be Haar system preserving, and show that such a morphism induces (via pullback) a *-morphism between the corresponding convolution algebras. We proceed to provide a plethora of examples of Haar system preserving morphisms and discuss connections to noncommutative CW-complexes and interval algebras. We prove that an inverse system of groupoids with Haar system preserving bonding maps has a limit, and that we get a corresponding direct system of groupoid $C^*$-algebras. An explicit construction of an inverse system of …
We analyse certain Haar systems associated to groupoids obtained by certain natural equivalence relations of dynamical nature on sets like $\{1,2,...,d\}^\mathbb{Z}$, $\{1,2,...,d\}^\mathbb{N}$, $S^1\times S^1$, or $(S^1)^\mathbb{N}$, where $S^1$ is the unitary circle. We also describe properties of transverse functions, quasi-invariant probabilities and KMS states for some examples of von Neumann algebras (and also $C^*$-Algebras) associated to these groupoids. We relate some of these KMS states with Gibbs states of Thermodynamic Formalism. While presenting new results, we will also describe …
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.
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.
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.
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.
Let $(G,α)$ and $(H,β)$ be locally compact groupoids with Haar systems. We define a topological correspondence from $(G,α)$ to $(H,β)$ to be a $G$-$H$-bispace $X$ on which $H$ acts properly and $X$ carries a continuous family of measures which is $H$-invariant and each measure in the family is $G$-quasi invariant. We show that a topological correspondence produces a $C^*$-correspondence from $C^*(G,α)$ to $C^*(H,β)$. We give many examples of topological correspondences.
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.
Let $G$ be a second countable, locally compact groupoid with Haar system, and let $\mathcal{A}$ be a bundle of $C^{\ast}$-algebras defined over the unit space of $G$ on which $G$ acts continuously. We determine conditions under which the associated crossed product $C^{\ast}(G;\mathcal{A})$ is a continuous trace $C^{\ast}$-algebra.
Von Neumann algebras as reduced twisted groupoid $C^*$-algebras
本文刻画了同构于约化扭曲群胚$C^*$-代数的冯·诺依曼代数，证明其为次齐次代数，并引入受控传播障碍，排除$B(H)$及Calkin代数的嵌入。
Let G be a second-countable locally-compact Hausdorff groupoid with a Haar system, and let {x_n} be a sequence in the unit space of G. We show that the notions of strength of convergence of {x_n} in the orbit space and measure-theoretic accumulation along the orbits are equivalent ways of realising multiplicity numbers associated to a sequence of induced representation of the groupoid C*-algebra.
Let E be a second-countable, locally compact, Hausdorff groupoid equipped with an action of T such that G:=E/T is a principal groupoid with Haar system λ. The twisted groupoid C*-algebra C*(E;G,λ) is a quotient of the C*-algebra of E obtained by completing the space of T-equivariant functions on E. We show that C*(E;G,λ) is postliminal if and only if the orbit space of G is T_0 and that C*(E;G, λ) is liminal if and only if the orbit space is T_1. We also show that C*(E;G, λ) has bounded trace if and only if G is integrable and that C*(E;G, λ) is a Fell algebra if and only if G is Cartan. Let \…
Suppose $G$ is a second countable, locally compact, Hausdorff groupoid with a fixed left Haar system. Let $\go/G$ denote the orbit space of $G$ and $\cs(G)$ denote the groupoid $C^*$-algebra. Suppose that $G$ is a principal groupoid. We show that $C^*(G)$ is CCR if and only if $\go/G$ is a $T_1$ topological space, and that $C^*(G)$ is GCR if and only if $\go/G$ is a $T_0$ topological space. We also show that $C^*(G)$ is a Fell Algebra if and only if $G$ is a Cartan groupoid.
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。
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 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 …
Suppose $G$ is a second countable, locally compact, Hausdorff groupoid with a fixed left Haar system. Let $\go/G$ denote the orbit space of $G$ and $C^*(G)$ denote the groupoid $C^*$-algebra. Suppose that the isotropy groups of $G$ are amenable. We show that $C^*(G)$ is CCR if and only if $\go/G$ is a $T_1$ topological space and all of the isotropy groups are CCR. We also show that $C^*(G)$ is GCR if and only if $\go/G$ is a $T_0$ topological space and all of the isotropy groups are GCR.
We introduce the concept of Roe C*-algebra for a locally compact groupoid whose unit space is in general not compact, and that is equipped with an appropriate coarse structure and Haar system. Using Connes' tangent groupoid method, we introduce an analytic index for an elliptic differential operator on a Lie groupoid equipped with additional metric structure, which takes values in the K-theory of the Roe C*-algebra. We apply our theory to derive a Lichnerowicz type vanishing result for foliations on open manifolds.
Suppose $G$ is a second countable, locally compact Hausdorff groupoid with abelian stabilizer subgroups and a Haar system. We provide necessary and sufficient conditions for the groupoid $C^*$-algebra to have Hausdorff spectrum. In particular we show that the spectrum of $C^*(G)$ is Hausdorff if and only if the stabilizers vary continuously with respect to the Fell topology, the orbit space $G^{(0)}/G$ is Hausdorff, and, given convergent sequences $χ_i\to χ$ and $γ_i\cdotχ_i \to ω$ in the dual stabilizer groupoid $\hat{S}$ where the $γ_i\in G$ act via conjugation, if $χ$ and $ω$ are elements o…
We discuss the relative K-theory for a $C^{*}$-algebra, $A$, together with a $C^{*}$-subalgebra, $A' \subseteq A$. The relative group is denoted $K_{i}(A';A), i = 0, 1$, and is due to Karoubi. We present a situation of two pairs $A' \subseteq A$ and $B' \subseteq B$ are related so that there is a natural isomorphism between their respective relative K-theories. We also discuss applications to the case where $A$ and $B$ are $C^{*}$-algebras of a pair of locally compact, Hausdorff topological groupoids, with Haar systems.
Let $(G,α)$ and $(H,β)$ be locally compact Hausdorff groupoids with Haar systems, and let $(X,λ)$ be a topological correspondence from $(G,α)$ to $(H,β)$ which induce the ${C}^*$-correspondence $\mathcal{H}(X)\colon {C}^*(G,α)\to {C}^*(H,β)$. We give sufficient topological conditions which when satisfied the ${C}^*$-correspondence $\mathcal{H}(X)$ is proper, that is, the ${C}^*$-algebra ${C}^*(G,α)$ acts on the Hilbert ${C}^*(H,β)$-module ${H}(X)$ via the comapct operators. Thus a proper topological correspondence produces an element in ${KK}({C}^*(G,α),{C}^*(H,β))$.
A theorem of Muhly-Renault-Williams states that if two locally compact groupoids with Haar system are Morita equivalent, then their associated convolution C*-algebras are strongly Morita equivalent. We give a new proof of this theorem for Lie groupoids. Subsequently, we prove a counterpart of this theorem in Poisson geometry: If two Morita equivalent Lie groupoids are s-connected and s-simply connected, then their associated Poisson manifolds (viz. the dual bundles to their Lie algebroids) are Morita equivalent in the sense of P. Xu.
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.
In this paper we refine a version of bivariant $K$-theory developed by Cuntz to define symmetric spectra representing the $KK$-theory of $C^\ast$-categories and discrete groupoid $C^\ast$-algebras. In both cases, the Kasparov product can be expressed as a smash product of spectra.
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 study Haar unitary random matrices with permuted entries. For a sequence of permutations $\left(σ_N\right)_N$, where $σ_N$ acts on $N\times N$ matrices we identify conditions under which the $\ast$--distribution of permuted Haar unitary matrices $U_N^{σ_N}$ is asymptotically circular and free from the unpermuted sequence $U_N$. We show that this convergence takes place in the almost sure sense. Moreover we show that our conditions on the sequence of permutations are generic in the sense that are almost surely satisfied by a sequence of random permutations.
Metrics on $C^{\ast}$-algebras of Étale groupoids from length functions
该研究探讨étale群胚长度函数导出的Dirac型算子在约化$C^{\ast}$-代数态空间上的度量性质，证明紧单位空间下生成伪度量，快速衰减变换群胚态空间分解为有限直径纤维度量空间，有限单位空间时纤维度量化弱*拓扑。
Metric dimension of $C^{\ast}$-algebras of cocycle twisted transformation groupoids: Growth and dynamical complexity
研究通过分层cLip-范数在扭变变换群胚C*-代数上构造CQMS结构，证明了当群为多项式增长时度量维数有上界，指数增长时维数无穷，揭示了群增长二分法在余圈扭变下的保持性。
If $G$ is a locally compact groupoid with a Haar system $λ$, then a positive definite function $p$ on $G$ has a form $p(x)=< L(x)ξ(d(x)),ξ(r(x))>$, where $L$ is a representation of $G$ on a Hilbert bundle ${\h}=(G^0,\{H_u\},μ)$, $μ$ is a quasi invariant measure on $G^0$ and $ξ\in L^{\infty}(G^0,{\h})$. [10]. In this paper firt we prove that if $μ$ is a quasi invariant ergodic measure on $G^0$, then two corresponding representations of $G$ and $C_c(G)$ are irreducible in the same time. Then by using the theory of positive linear functionals on $C^*(G)$ we show that when $μ$ is an ergodic quasi …
Invariant measures and traces on groupoid $\boldsymbol{\mathrm{C}^\ast}$-algebras
本文研究了群体本质$\mathrm{C}^\ast$-代数上迹的存在条件，并证明了有限状态自相似群的 gauge-invariant 代数具有唯一的迹状态。
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 $-Ruelle triples and c…
We establish that a second countable locally compact groupoid possessing a continuous Haar system is topologically amenable if and only if it is Borel amenable. We give some examples and applications.
The Unitary Conjugation Groupoid of a Type I C*-Algebra: Topology, Fell Continuity, and the Canonical Diagonal Embedding
本文提出了一种基于II型C*-代数的规范群oid构造，通过规范对角嵌入将代数嵌入到群oidC*-代数中，展示了其在K-理论和拓扑学中的应用。
A note on projections in étale groupoid algebras and diagonal preserving homomorphisms
本文刻画了满足“豪斯多夫充足广群代数的任意$\ast$-同态均自动保对角”性质的环，其恰好是Carlsen研究的复数共轭闭子环，且给出了比Leavitt路代数情形更简洁的广群层面证明。
Isomorphisms in K-theory from isomorphisms in groupoid homology theories
该研究证明无挠顺从充裕群胚中由平展对应诱导的群胚同调同构可导出相关$\text{C}^\text{*}$-代数K-理论同构，借此推广了X. Li的左正则逆半群$\text{C}^\text{*}$-代数K-理论公式，核心方法是发展ABC谱序列的函子性。
The bulk-boundary and a new bulk-defect correspondence principles are formulated using groupoid algebras. The new strategy relies on the observation that the groupoids of lattices with boundaries or defects display spaces of units with invariant accumulation manifolds, hence they can be naturally split into disjoint unions of open and closed invariant sub-sets. This leads to standard exact sequences of groupoid $C^\ast$-algebras that can be used to associate a Kasparov element to a lattice defect and to formulate an extremely general bulk-defect correspondence principle. As an application, we …
We consider the algebra $\dotΣ(\mathcal L)$ generated by the inner-limit derivations over the ${\rm GICAR}$ algebra of a fermion gas populating an aperiodic Delone set $\mathcal L$. Under standard physical assumptions such as finite interaction range, Galilean invariance and continuity with respect to the aperiodic lattice, we demonstrate that the image of $\dot Σ(\mathcal L)$ through the Fock representation can be completed to a groupoid-solvable pro-$C^\ast$-algebra. Our result is the first step towards unlocking the $K$-theoretic tools available for separable $C^\ast$-algebra for applicatio…
We construct a new class of finite-dimensional C^*-quantum groupoids at roots of unity q=e^{iπ/\ell}, with limit the discrete dual of the classical SU(N) for large orders. The representation category of our groupoid turns out to be tensor equivalent to the well known quotient C^*-category of the category of tilting modules of the non-semisimple quantum group U_q({\mathfrak sl}_N) of Drinfeld, Jimbo and Lusztig. As an algebra, the C^*-groupoid is a quotient of U_q({\mathfrak sl}_N). As a coalgebra, it naturally reflects the categorical quotient construction. In particular, it is not coassociati…
Inspired by the work of Paterson on $C^{\ast}$-algebras of directed graphs, we show how to associate a groupoid $\mathfrak{G}_{\mathcal{G}}$ to an ultragraph $\mathcal{G}$ in such a way that the $C^*$-algebra of $\mathfrak{G}_{\mathcal{G}}$ is canonically isomorphic to Tomforde's $C^*$-algebra $C^*(\mathcal{G})$. The groupoid $\mathfrak{G}_{\mathcal{G}}$ is built from an inverse semigroup $S_{\mathcal{G}}$ naturally associated to $\mathcal{G}$.
We review various notions of correspondences for locally compact groupoids with Haar systems, in particular a recent definition due to R.D. Holkar. We give the construction of the representations induced by such a correspondence. Finally, we extend the construction of induced representations to hypergroupoids.
In the previous article, we proved that a topological correspondence $(X,λ)$ from a locally compact groupoid with a Haar system $(G,α)$ to another one, $(H,β)$, produces a $C^*$-correspondence $\mathcal{H}(X)$ from $C^*(G,α)$ to $C^*(H,β)$. In the present article, we describe how to form a composite of two topological correspondences when the bispaces are Hausdorff and second countable in addition to being locally compact.
Homology and K-theory for self-similar actions of groups and groupoids
本文为自相似群作用构造长正合序列，计算其群胚同调和C*-代数K-理论，并证明包含Grigorchuk群的Röver单群有理无环但Schur乘子非平凡。
Synchronizing Dynamical Systems: their groupoids and $C^*$-algebras
本文提出同步动力系统以推广 Smale 空间，并研究其相关 $C^\ast$-代数，证明其同宿代数含具有类似 Smale 空间同宿代数行为的理想。
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。
Non-isomorphism of reduced free group $C^\ast$-algebras
本文通过新方法证明了不同自由群的简化解$C^\ast$-代数非同构，恢复了Pimsner和Voiculescu的经典结果。
Simplicity of $C^*$-algebras of contracting self-similar groups
本文证明了收缩自相似群对应的$C^*$-代数与复$\t$-代数单性等价，并改进了判定算法，为充足群胚提供了新实例。
Self-adjoint traces on the Pedersen ideal of $\mathrm{C}^\ast$-algebras
本文系统研究 $\mathrm{C}^\ast$-代数 Pedersen 理想上的自伴迹集合 $T_{\mathbb R}(A)$，建立其 Kadison 对偶，计算主扭平展群胚代数的该集合，并对一大类代数正面回答迹问题。
$K$-theory of two-dimensional substitution tiling spaces from $AF$-algebras
本文证明在合理假设下，二维替换铺嵌空间不稳定广群 $C^\ast$-代数的 $K$-理论可由替换规则及其 $1$-骨架类似物的 $AF$-代数 $K$-理论显式重构，通过相对 $K$-理论与切除转化为纯拓扑正合序列，并给出 Sage 计算示例。
We compute the $K$-theory groups of the groupoid C$^\ast$-algebra of the chair tiling, using a new method. We use exact sequences of Putnam to compute these groups from the $K$-theory groups of the $AF$-algebras of the substitution and the induced lower dimensional substitutions on edges and vertices.
Finite free position of maximal abelian $\ast$-subalgebras of the matrix algebra
本文研究矩阵代数中极大交换$\ast$-子代数的有限自由位置，通过行列式条件刻画其成立当且仅当$n\le3$且$\sqrt{n}U$为复Hadamard矩阵，并引入均匀主子式偏差量化$n\ge4$时的偏离，给出对称性、单调性及下界。
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