We construct an algebra of pseudodifferential operators on each groupoid in a class that generalizes differentiable groupoids to allow manifolds with corners. We show that this construction encompasses many examples. The subalgebra of regularizing operators is identified with the smooth algebra of the groupoid, in the sense of non-commutative geometry. Symbol calculus for our algebra lies in th...

In 1949, M. Sholander [4] showed that every medial cancellation groupoid can be embedded into a medial quasigroup. In this note we prove the dual assertion, that every medial division groupoid is a homomorphic image of a medial quasigroup.

In this paper we study Poisson actions of complete Poisson groups, without any connectivity assumption or requiring the existence of a momentum map. For any complete Poisson group G with dual G⋆ we obtain a suitably connected integrating symplectic double groupoid S. As a consequence, the cotangent lift of a Poisson action on an integrable Poisson manifold P can be integrated to a Poisson actio...

We present a number of findings concerning groupoid dynamical systems and groupoid crossed products. The primary result is an identification of the spectrum of the groupoid crossed product when the groupoid has continuously varying abelian stabilizers and a well behaved orbit space. In this case, the spectrum of the crossed product is homeomorphic, via an induction map, to a quotient of the spe...

We prove the existence of a strict deformation quantization for the canonical Poisson structure on the dual of an integrable Lie algebroid. It follows that any Lie groupoid C-algebra may be regarded as a result of a quantization procedure. The C-algebra of the tangent groupoid of a given Lie groupoid G (with Lie algebra G) is the C-algebra of a continuous field of C-algebras over R with fibers ...

We study certain function algebras and their operator algebra completions on r-discrete abelian groupoids, the corresponding conditional expectations, maximal abelian subalgebras (masa) and eigen-functionals. We give a semidirect product decomposition for an abelian groupoid. This is done through a matched pair and leads to a C*-diagonal (for a special case). We use this decomposition to study ...

We propose a definition of compact quantum groupoids in the setting of C -algebras, associate to such a quantum groupoid a regular C -pseudo-multiplicative unitary, and use this unitary to construct a dual Hopf C -bimodule and to pass to a measurable quantum groupoid in the sense of Enock and Lesieur. Moreover, we discuss examples related to compact and to étale groupoids and study principal co...

In ([L2]), Franck Lesieur had introduced a notion of measured quantum groupoid, in the setting of von Neumann algebras. In an annex of [E6], Lesieur’s axioms have been simplified. In this article, we suppose that the basis is central; in that case, we prove that a specific sub-C∗ algebra is, in a sense, invariant under all the data which define the measured quantum group, which allow us to prov...

Given a normal subgroup bundle A of the isotropy groupoid ?, we obtain twisted action quotient ?/A on group C?-algebras determined by whose crossed product recovers C?-algebra C?(?). Restricting to case where is abelian, describe C?(?) as associated T-groupoid over tranformation obtained from canonical Pontryagin dual space A. We give some illustrative examples this result.