Generalized Topological Groupoids

Our aim in this paper is to give the notion of generalized topological groupoid which is a generalization of the topological groupoid by using the notion of generalized topology defined by Csasz ´ ar [6]. We in- ´ vestigate the basic facts in the groupoid theory in terms of generalized topological groupoids. We present the action of a generalized topological groupoid on a generalized topological space. We obtain some characterizations about this concept that is called the generalized topological action. Beside these, we give definition of a generalized topological crossed module by generalizing the concept of crossed module defined on topological groupoids. At the last part of the study, we show how a generalized topological crossed module can be obtained from a generalized topological groupoid and how a generalized topological groupoid can be obtained from a generalized topological crossed module.

NON-COMMUTATIVE STONE DUALITY: INVERSE SEMIGROUPS, TOPOLOGICAL GROUPOIDS AND C*-ALGEBRAS

We study a non-commutative generalization of Stone duality that connects a class of inverse semigroups, called Boolean inverse ∧-semigroups, with a class of topological groupoids, called Hausdorff Boolean groupoids. Much of the paper is given over to showing that Boolean inverse ∧-semigroups arise as completions of inverse ∧-semigroups we call pre-Boolean. An inverse ∧-semigroup is pre-Boolean if and only if every tight filter is an ultrafilter, where tight filters are defined by combining ideas of both Exel and Lenz. A simple necessary condition for a semigroup to be pre-Boolean is derived and a variety of examples of inverse semigroups are shown to satisfy it. Thus the polycyclic inverse monoids, and certain Rees matrix semigroups over the polycyclics, are pre-Boolean and it is proved that the groups of units of their completions are precisely the Thompson–Higman groups Gn, r. The inverse semigroups arising from suitable directed graphs are also pre-Boolean and the topological groupoids arising from these graph inverse semigroups under our non-commutative Stone duality are the groupoids that arise from the Cuntz–Krieger C*-algebras. An elementary application of our theory shows that the finite, fundamental Boolean inverse ∧-semigroups are just the finite direct products of finite symmetric inverse monoids. Finally, we explain how tight filters are related to prime filters setting the scene for future work.

Restricted Algebras on Inverse Semigroups—Part II: Positive Definite Functions

The relation between representations and positive definite functions is a key concept in harmonic analysis on topological groups. Recently this relation has been studied on topological groupoids. In this paper, we investigate the concept of restricted positive definite functions and their relation with restricted representations of an inverse semigroup. We also introduce the restricted Fourier and Fourier-Stieltjes algebras of an inverse semigroup and study their relation with the corresponding algebras on the associated groupoid.

Imprimitivity theorem for groupoid representations

We define and investigate the concept of the groupoid representation induced by a representation of the isotropy subgroupoid. Groupoids in question are locally compact transitive topological groupoids. We formulate and prove the imprimitivity theorem for such representations which is a generalization of the classical Mackey’s theorem known from the theory of group representations.