The type I and CCR properties forgroupoids and inverse semigroups

Abstract: This licentiate thesis consists of one paper about unitary representationtheory of ample groupoids and semigroups together with generalizationsto étale and non-Hausdorff groupoids. In the paper we study algebraically the type I and CCR properties forample Hausdorff groupoids. Clarke and Van Wyk proved that both ofthese properties admit a topological characterization for Hausdorff second countable groupoids in terms of separation properties of their orbitspace and the isotropy groups. Using a Stone type duality between ample groupoids and Boolean inverse semigroups with meets, we exploit thischaracterization to get a purely algebraic statement. We also apply thoseresults to get characterizations of the type I and CCR properties for inverse semigroups using their Boolean inverse completions. The generalization is about characterizing the same properties for both étale and ample non-necessarily Hausdorff groupoids which nonethelesshave Hausdorff unit spaces. In this setup, we first give a direct proofof the topological characterization for the CCR property which doesn't rely on the disintegration theory. The argument cannot be adapted toget an easier proof in the type I case, but we rather explain how to geta proof following the original ideas of Clark and Van Wyk in that case.Finally, we state for both étale and ample groupoids algebraic conditionsequivalent to the CCR and GCR properties on their pseudogroup of openand compact open bisections respectively.

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