Groupoid Actions Research Articles

The unitary implementation of a measured quantum groupoid action

Frank Lesieur a introduit une notion de groupoide quantique mesure, dans le cadre des algebres de von Neumann, en s'inspirant des groupes quantiques localement compacts de Kustermans et Vaes (dans la version de cette construction faite dans le cadre des algebres de von Neumann). Dans un article precedent, l'auteur a introduit les notions d'action, de produit croise, d'action duale d'un groupoide quantique mesure ; un theoreme de bidulaite des actions a ete demontre. Cet article continue ce programme: nous demontrons l'existence d'une implementation standard d'une action, et un theoreme de bidulaite pour les poids. Sont ainsi generalises des resultats qui avaient ete demontres par S. Vaes pour les groupes quantiques localement compacts, et par T. Yamanouchi pour les groupoides mesures.

Fractal Property in B(H) Induced by Partial Isometries

We consider the groupoid C*-dynamical systems \({(A,{\mathbb G},\alpha ),}\) where A is a C*-algebra in \({B(H),{\mathbb G}}\) is a certain groupoid, and α is an embedding groupoid action of \({{\mathbb G},}\) acting on A in B(H). In particular, we are interested in the case where the groupoid \({{\mathbb G}}\) is a fractaloid, a groupoid with fractal property. Moreover, we restrict our interests to the case where \({{\mathbb G}}\) is generated by finitely many partial isometries in B(H). We observe the basic properties of such C*-dynamical systems.

Tannaka duality for proper Lie groupoids

By replacing the category of smooth vector bundles of finite rank over a manifold with the category of what we call smooth Euclidean fields, which is a proper enlargement of the former, and by considering smooth actions of Lie groupoids on smooth Euclidean fields, we are able to prove a Tannaka duality theorem for proper Lie groupoids. The notion of smooth Euclidean field we introduce here is the smooth, finite dimensional analogue of the usual notion of continuous Hilbert field.

Small worlds and Red Queens in the Global Workspace: An information-theoretic approach

Networks of small worlds and Red Queen dynamic structures as analytic-descriptive methods are ubiquitous in expanding areas of research in the cognitive neurosciences, sociopsychological systems and parallel machine cognition. In the setting of the Baars Global Workspace theory, we apply a semantically oriented information-theoretic mechanism, making use of the Onsager relations of statistical physics and the geometry of associated networks. For such networks attention is paid to the concept of dynamic groupoids, their symmetries and other properties. In the case of structural networks influenced by the orbit equivalence relations of a groupoid action, we describe how crosstalk, the emergence of subnetwork giant components and noise induced opportunities contribute to the structure of small world networks and Red Queen dynamics. We create some dynamic models and show how these systems can be formally encapsulated within the idea of a groupoid atlas. Further, we utilize specific geometrical methods to describe the formation of geodesics in the corresponding networks, and include the holonomy concept as providing a geometric representation associated with a phase transition.

Automorphisms and Homotopies of Groupoids and Crossed Modules

This paper is concerned with the algebraic structure of groupoids and crossed modules of groupoids. We describe the group structure of the automorphism group of a finite connected groupoid C as a quotient of a semidirect product. We pay particular attention to the conjugation automorphisms of C, and use these to define a new notion of groupoid action. We then show that the automorphism group of a crossed module of groupoids \(\mathcal{C}\), in the case when the range groupoid is connected and the source group totally disconnected, may be determined from that of the crossed module of groups \(\mathcal{C}_u\) formed by restricting to a single object u. Finally, we show that the group of homotopies of \(\mathcal{C}\) may be determined once the group of regular derivations of \(\mathcal{C}_u\) is known.

Vertex-Compressed Subalgebras of a Graph von Neumann Algebra

In Cho (Acta Appl. Math. 95:95–134, 2007 and Complex Anal. Oper. Theory 1:367–398, 2007), we introduced Graph von Neumann Algebras which are the (groupoid) crossed product algebras of von Neumann algebras and graph groupoids via graph-representations, which are groupoid actions. In Cho (Acta Appl. Math. 95:95–134, 2007), we showed that such crossed product algebras have the amalgamated reduced free probabilistic properties, where the reduction is totally depending on given directed graphs. Moreover, in Cho (Complex Anal. Oper. Theory 1:367–398, 2007), we characterize each amalgamated free blocks of graph von Neumann algebras: we showed that they are characterized by the well-known von Neumann algebras: Classical group crossed product algebras and (operator-valued) matricial algebras. This shows that we can provide a nicer way to investigate such groupoid crossed product algebras, since we only need to concentrate on studying graph groupoids and characterized algebras. How about the compressed subalgebras of them? i.e., how about the inner (cornered) structures of a graph von Neumann algebra? In this paper, we will provides the answer of this question. Consequently, we show that vertex-compressed subalgebras of a graph von Neumann algebra are characterized by other graph von Neumann algebras. This gives the full characterization of the vertex-compressed subalgebras of a graph von Neumann algebra, by other graph von Neumann algebras.

Operator Algebraic Quotient Structures Induced by Directed Graphs

Our result is about inclusions for (finite or infinite) countable directed graphs. In the proof, we use Free Probability Theory, groupoids, and algebras of operators (von Neumann algebras). We show that inclusions of directed graphs induce quotients for associated groupoid actions. With the use of operator thechniques, we then establish a duality between inclusions of graphs on the one hand and quotients of algebras on the other. Our main result is that each connected graph induces a quotient generated by a free group. This is a generalization of the notion of induced representations in the context of unitary representations of groups, i.e., the induction from the representations of a subgroup of an ambient group. The analogue is to systems of imprimitivity based on the homogeneous space. The parallel of this is the more general context of graphs (extending from groups to groupoids): We first prove that inclusions for connected graphs correspond to free group quotients, and we then build up the general case via connected components of given graphs.

Gravitational Aharonov-Bohm Effect

We study a purely gravitational Aharonov-Bohm effect. The space-time curvature is concentrated in the quasiregular singularity of a cosmic string, outside of which space-time is (locally) flat. The symmetries of this field configuration are described by the groupoid symmetries rather than by the usual group symmetries. The groupoid in question is formed by homotopy classes of piecewise smooth paths in the cosmic string region. A gravitational counterpart of the Aharonov-Bohm effect occurs if the symmetry of the system, with respect to the groupoid action, is broken down.

Characterization of Amalgamated Free Blocks of a Graph von Neumann Algebra

We identify two noncommutative structures naturally associated with countable directed graphs. They are formulated in the language of oper- ators on Hilbert spaces. If G is a countable directed graphs with its vertex set V (G) and its edge set E(G), then we associate partial isometries to the edges in E(G) and projections to the vertices in V (G). We construct a correspond- ing von Neumann algebra MG as a groupoid crossed product algebra M ×α G of an arbitrary fixed von Neumann algebra M and the graph groupoid G induced by G, via a graph-representation (or a groupoid action) α. Graph groupoids are well-determined (categorial) groupoids. The graph groupoid G of G has its binary operation, called admissibility. This G has concrete local parts Ge, for all e ∈ E(G). We characterize Me = M ×α Ge of MG, induced by the local parts Ge of G, for all e ∈ E(G). We then characterize all amal- gamated free blocks Me = vN(Me, DG )o fMG. They are chracterized by well-known von Neumann algebras: the classical group crossed product alge- bras M ×λ(e)Z, and certain subalgebras M α e 2 (M ) of operator-valued matricial algebra M ⊗C M2(C). This shows that graph von Neumann algebras identify the key properties of graph groupoids. Mathematics Subject Classification (2000). 47L99.

Amenability and the Liouville property

We present a new approach to the amenability of groupoids (both in the measure theoretical and the topological setups) based on using Markov operators. We introduce the notion of an invariant Markov operator on a groupoid and show that the Liouville property (absence of non-trivial bounded harmonic functions) for such an operator implies amenability of the groupoid. Moreover, the groupoid action on the Poisson boundary of any invariant operator is always amenable. This approach subsumes as particular cases numerous earlier results on amenability for groups, actions, equivalence relations and foliations. For instance, we establish in a unified way topological amenability of the boundary action for isometry groups of Gromov hyperbolic spaces, Riemannian symmetric spaces and affine buildings.

Contact reduction and groupoid actions

We introduce a new method to perform reduction of contact manifolds that extends Willett’s and Albert’s results. To carry out our reduction procedure all we need is a complete Jacobi map J : M → Γ 0 J:M \rightarrow \Gamma _0 from a contact manifold to a Jacobi manifold. This naturally generates the action of the contact groupoid of Γ 0 \Gamma _0 on M M , and we show that the quotients of fibers J − 1 ( x ) J^{-1}(x) by suitable Lie subgroups Γ x \Gamma _x are either contact or locally conformal symplectic manifolds with structures induced by the one on M M . We show that Willett’s reduced spaces are prequantizations of our reduced spaces; hence the former are completely determined by the latter. Since a symplectic manifold is prequantizable iff the symplectic form is integral, this explains why Willett’s reduction can be performed only at distinguished points. As an application we obtain Kostant’s prequantizations of coadjoint orbits. Finally we present several examples where we obtain classical contact manifolds as reduced spaces.

The Equivariant Analytic Index for Proper Groupoid Actions

The paper constructs the analytic index for an elliptic pseudodifferential family of $L^{m}_{\rho,\de}$-operators invariant under the proper action of a continuous family groupoid on a $G$-compact, $C^{\infty,0}$ $G$-space.

Non-commutative coordinate rings and stacks

Let $Y \rightrightarrows X$ be a finite flat groupoid scheme with $X$ a quasi-projective variety and let $S$ be its coarse moduli scheme. We associate to the groupoid scheme a coherent sheaf of algebras $\mathcal{O}_{X / Y}$ on $S$ which we call the non-commutative coordinate ring of the groupoid scheme. We show that when $X$ is a smooth curve and the groupoid action is generically free, the non-commutative coordinate rings which can occur are, up to Morita equivalence, the hereditary orders on smooth curves. This gives a bijective correspondence between smooth one-dimensional Deligne?Mumford stacks of finite type and Morita equivalence classes of hereditary orders on smooth curves.

Local index theory over étale groupoids

We give a superconnection proof of Connes' index theorem for proper cocompact actions of etale groupoids. This includes Connes' general foliation index theo- rem for foliations with Hausdorholonomy groupoid.

A Galois Correspondence for II1 Factors and Quantum Groupoids

We establish a Galois correspondence for finite quantum groupoid actions on II1 factors and show that every finite index and finite depth subfactor is an intermediate subalgebra of a quantum groupoid crossed product. Moreover, any such subfactor is completely and canonically determined by a quantum groupoid and its coideal *-subalgebra. This allows us 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 finite-dimensional C*-algebras related to it.

A Unified Approach to Poisson Reduction

Given any Poisson action G×P→P of a Poisson–Lie group G we construct an object Ω=T*G*T*P which has both a Lie groupoid structure and a Lie algebroid structure and which is a half-integrated form of the matched pair of Lie algebroids which J.-H. Lu associated to a Poisson action in her development of Drinfeld's classification of Poisson homogeneous spaces. We use Ω to give a general reduction procedure for Poisson group actions, which applies in cases where a moment map in the usual sense does not exist. The same method may be applied to actions of symplectic groupoids and, most generally, to actions of Poisson groupoids.

Using Rewriting Systems to Compute Left Kan Extensions and Induced Actions of Categories

The aim is to apply string-rewriting methods to compute left Kan extensions, or, equivalently, induced actions of monoids, categories, groups or groupoids. This allows rewriting methods to be applied to a greater range of situations and examples than before. The data for the rewriting is called a Kan extension presentation. The paper has its origins in earlier work by Carmody and Walters who gave an algorithm for computing left Kan extensions based on extending the Todd–Coxeter procedure, an algorithm only applicable when the induced action is finite. The current work, in contrast, gives information even when the induced action is infinite.

Fixed point algebras of groupoid actions and coactions on von Neumann algebras

Fixed point algebras of groupoid actions and coactions on von Neumann algebras

Duality for actions and coactions of measured groupoids on von Neumann algebras

Relative tensor products of Hilbert spaces over abelian von Neumann algebras Coproducts of groupoid von Neumann algebras Actions and coactions of measured groupoids on von Neumann algebras Crossed products by groupoid actions and their dual coactions Crossed products by groupoid coactions and their dual actions Duality for actions on von Neumann algebras Duality for integrable coactions on von Neumann algebras Examples of actions and coactions of measured groupoids on von Neumann algebras.

Dual Weights on Crossed Products by Groupoid Actions

In [Y2], we carried out the program of realizing the crossed product by a groupoid action as the left von Neumann algebra of a left Hilbert algebra naturally attached to the given covariant system. As a consequence, it was shown, as in the case of group actions, that, for each faithful normal positive functional q) on the algebra on which the groupoid is acting, there always exists a faithful normal semifinite cp on the crossed product, called the dual of (JP. (To avoid difficulty, we dealt with a positive functional only, not with a weight). Several expected results were established such as the fact that the modular automorphism group of q> extends that of cp. It is naturally expected at this stage that this dual construction could be done also by exhibiting an operator valued of the crossed product to the original algebra, as Haagerup showed in [H4]. The purpose of this paper is to show that this philosophy is indeed the case. The main strategy to achieye our goal can be found in [H4]. However, since we know little about Fourier analysis (or harmonic analysis) on a measured groupoid, we need to provide ourselves with relevant information in this direction. For example, to the best of author's knowledge, no one has ever intensively studied the Plancherel weight of a measured groupoid. Thus, naturally, little is known about what functions must be qualified to be called positive definite. Moreover, it should be remarked that the unitary operator A(y) itself, where A(-) is the regular representation of a measured groupoid % has no meaning in the crossed product M x^ by an action a of (it is not even a member of M x^), while, in the group case, it is a typical kind of operators that generate the crossed product. This fact makes our argument more difficult than the one in the case of group actions. For example, because of this situation, we need to adopt an approach in §4 which is different from, but still as interesting as, the ones taken in [H4] or [E&S].