## The structure of GCR and CCR groupoid C*-algebras

**Author: **van Wyk, Daniel Willem

**Date:** 2017

**Publisher: **University of Otago

**Type: **Thesis

**Link to this item using this URL: **http://hdl.handle.net/10523/7583

### Abstract

We remove the assumptions of amenability in two theorems of Clark about C*-algebras of locally compact groupoids. The first result is that if the groupoid C*-algebra is GCR, or equivalently then the groupoid's orbits are locally closed. We prove the contrapositive. We begin by constructing a direct integral representation of the groupoid C*-algebra with respect to a measure on the groupoid's unit space. If the orbits are not locally closed, then there is a non-trivial ergodic measure on the unit space. We adapt a known result for transformation groups to groupoids, which shows that the direct integral representation cannot be type I if the measure on the unit space is non-trivially ergodic. The second result is that if the groupoid C*-algebra is CCR, then the groupoid's orbits are closed. Here we show that if a representation of a stability subgroup is induced to a representation of the groupoid C*-algebra, then the induced representation is equivalent to a representation as multiplication operators acting on a vector-valued L2-space. If we assume the groupoid C*-algebra is CCR, but an orbit is not closed, then the equivalence of two representations as multiplication operators leads to a contradiction.

**Subjects: **C*-algebras, Groupoids, Operator Algebras, GCR, CCR

**Citation:** ["van Wyk, D. W. (2017). The structure of GCR and CCR groupoid C*-algebras (Thesis, Doctor of Philosophy). University of Otago. Retrieved from http://hdl.handle.net/10523/7583"]

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