Arlan Ramsay
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.
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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.
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.
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.
Given a locally convex space $(\mathcal{A},τ)$ with a Hausdorff locally convex topology $τ$ such that the following maps are continuous; $u \mapsto u^*$ for all $u \in \mathcal{A}$, $x \mapsto x\cdot y$ and $x \mapsto z\cdot x$ for every left and right multipliers of $\mathcal{A}$. In this paper we re-characterized the locally convex partial $*$-algebra $(\mathcal{A}, Γ,\cdot,*,τ)$ arising from these continuous maps in terms of convolution algebra of a Lie groupoid $Γ\rightrightarrows \mathcal{A}$. This is advantageous because the pathologies of the underlying spaces owing to their quantum mec…
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…
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。
Locally convex bialgebroid of an action Lie groupoid
本文提出了一种通过局部凸双李代数重构行动李群体的方法，并展示了其在谱行动李群体中的应用。
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 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).
Normal elements (or multipliers) of the C* algebra of a certain class of locally compact groupoids admit a natural faithful representation as normal operators on the $L^2$-space of a dense orbit of the groupoid. We prove norm estimates on the product between elements of the functional calculus of these operators and multiplication operators, subject to suitable restrictions expressed in terms of the orbit structure of the groupoid. As a consequence, one gets uniform estimates on the local behavior of the evolution group of the operators.
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.
Inverse limits of locally convex Lie groupoids
该研究探讨局部凸李群胚与李代数胚的严格逆系统，建立函子同构，给出可积巴拿赫-李代数胚逆系统的可积化条件，描述源单连通性障碍，并用射流群胚等实例阐释理论。
Continuous field of Orlicz space on locally compact groupoids and related results
研究局部紧广群上奥利奇空间连续场，引入\((L^{\Phi}_{0},\Delta_{1})\)，给出\(E^{\Phi}_{0}\)成巴拿赫代数的条件及\(E^{\Phi}_{0}\)子模成左理想的条件，还给出莫尔斯 - 特兰苏空间卷积子空间刻画的广群类似物。
We study bounded operators defined in terms of the regular representations of the $C^*$-algebra of an amenable, Hausdorff, second countable locally compact groupoid endowed with a continuous $2$-cocycle. We concentrate on spectral quantities associated to natural quotients of this twisted algebra, such as the essential spectrum, the essential numerical range, and Fredholm properties. We obtain decompositions for the regular representations associated to units of the groupoid belonging to a free locally closed orbit, in terms of spectral quantities attached to points (or orbits) in the boundary…
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。
On the restriction maps of the Fourier and Fourier-Stieltjes algebras over locally compact groupoids
本文将群上 Fourier 与 Fourier-Stieltjes 代数限制映射的满射性结果推广到局部紧广群，刻画 étale、传递及群丛等情形下的满射条件，并借助 FD 性质给出非满射例子及 Banach 代数性质缺失的条件。
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.
A category of locally convex Lie algebroids
研究一阶局部凸李代数胚，在巴斯蒂亚尼微积分背景下，定义相关层和态射证明其构成范畴，还研究表示等，表明局部凸李群胚有一阶李代数胚及态射，给出相关应用如证明流代数胚性质及李 II 定理。
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.
Dynamic asymptotic dimension growth for group actions and groupoids
该研究引入离散群作用及局部紧平展群胚的动态渐近维数增长概念，证明其与有界几何离散度量空间的渐近维数增长等价，推导相关群胚的可和性结论，并将其应用于证明带系数Baum-Connes猜想成立。
Weak containment of representation on topological groupoids
本文研究拓扑群胚连续酉表示的弱包含，证明诱导与内张量积保持该性质，引入拓扑不变平均并建立群胚版Greenleaf定理，还给出相关诱导表示限制的独立结果。
Measurable bounded cohomology of measured groupoids
本文为测度群胚引入可测有界上同调理论，证明其同构于局部紧群的连续有界上同调，并揭示在轨道等价与测度等价下的不变性及顺从群胚的平凡性。
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…
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