Groupoids graded by the groupoid of bijections between finite sets admit generating functions which encode cardinalities their components. As suggested in work Baez and Dolan, we use analytic continuation such to define a complex-valued cardinality for groupoids whose usual diverges. The complex nature invariant is shown reflect recursion structure refer as ‘nested equivalence’.

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...