Franck Lesieur had introduced in his thesis (now published in an expended and revised version in the 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 h...

If p:B→G is a Fell bundle over an étale groupoid, then we show that there norm reducing injective linear map j:Cr⁎(G;B)→Γ0(G;B) generalizing the well known j:Cr⁎(G)→C0(G) in case of groupoid.

Orbifold groupoids have been recently widely used to represent both effective and ineffective orbifolds. We show that every orbifold group-oid can be faithfully represented on a continuous family of finite dimensional Hilbert spaces. As a consequence we obtain the result that every orbifold groupoid is Morita equivalent to the translation groupoid of an almost free action of a proper bundle of ...

In this paper, we construct an explicit quasi-isomorphism to study the cyclic cohomology of a deformation quantization over a Riemannian étale groupoid. Such a quasi-isomorphism allows us to propose a general algebraic index problem for Riemannian étale groupoids. We discuss solutions to that index problem when the groupoid is proper or defined by a constant Dirac structure on a 3-dimensional t...

Many interesting C∗-algebras can be viewed as quantizations of Poisson manifolds. I propose that a Poisson manifold may be quantized by a twisted polarized convolution C∗-algebra of a symplectic groupoid. Toward this end, I define polarizations for Lie groupoids and sketch the construction of this algebra. A large number of examples show that this idea unifies previous geometric constructions, ...

We first describe a Rieffel induction system for groupoid crossed products. We then use this induction system to show that, given a regular groupoid G and a dynamical system (A,G,α), every irreducible representation of A⋊G is induced from a representation of the group crossed product A(u)⋊Su where u ∈ G(0), A(u) is a fibre of A, and Su is a stabilizer subgroup of G.

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 continuous representations of compact groupoids. We show that irreducible representationshave finite dimensional fibres. We prove the Schur’s lemma, Gelfand-Raikov theore...

We explore the connections between automata, groups, limit spaces of self-similar actions, and tilings. In particular, we show how a group acting “nicely” on a tree gives rise to a self-covering of a topological groupoid, and how the group can be reconstructed from the groupoid and its covering. The connection is via finite-state automata. These define decomposition rules, or self-similar tilin...

In this paper, we show how certain metabelian groups can be found within polynomial evaluation groupoids. We show that every finite abelian group can beobtained as a polynomial evaluation groupoid.