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.