# Gabriel Favre: The type I and CCR properties for groupoids and inverse semigroups

**Time: **
Thu 2022-01-20 15.00 - 17.00

**Location: **
Kräftriket, House 5, Room 14 and Zoom (Meeting ID: 627 2410 9446)

**Video link: **
Meeting ID: 627 2410 9446

**Doctoral student: **
Gabriel Favre

**Opponent: **
Fernando Lledó (Universidad Carlos III de Madrid)

**Supervisor: **
Sven Raum

### Abstract

This licentiate thesis consists of one paper about unitary representation theory of ample groupoids and semigroups together with generalizations to étale and non-Hausdorff groupoids.

In the paper we study algebraically the type I and CCR properties for ample Hausdorff groupoids. Clarke and Van Wyk proved that both of these properties admit a topological characterization for Hausdorff second countable groupoids in terms of separation properties of their orbit space and the isotropy groups. Using a Stone type duality between ample groupoids and Boolean inverse semigroups with meets, we exploit this characterization to get a purely algebraic statement. We also apply those results 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 nonetheless have Hausdorff unit spaces. In this setup, we first give a direct proof of the topological characterization for the CCR property which doesn't rely on the disintegration theory. The argument cannot be adapted to get an easier proof in the type I case, but we rather explain how to get a proof following the original ideas of Clark and Van Wyk in that case.

Finally, we state for both étale and ample groupoids algebraic conditions equivalent to the CCR and GCR properties on their pseudogroup of open and compact open bisections respectively.