A groupoid approach to C*-algebras

TL;DR: The maximal proper closed inverse submonoids of the polycyclic inverse monoids, also known as Cuntz inverse semigroups, are characterised and all their primitive partial permutation representations are determined.

TL;DR: In this article, the authors associate an r-discrete, locally compact groupoid, similar to the Cuntz-Krieger groupoid with a special embedding of circle algebras having the same spectrum.

TL;DR: In this paper, a generalization of Renault's disintegration theorem for representations of etale groupoids on Hilbert spaces has been proposed, where the authors establish a correspondence between contractive L^p-representations of an ε-groupoid G and tight regular Lǫp representations of any countable inverse semigroup of open slices of G that is a basis for the topology of G.

TL;DR: In this paper, the construction of weak Hopf algebras (or quantum groupoids) associated to a matched pair of groupoids and certain cocycle data is studied.