2 Lisa Orloff

Suppose G is a second countable, locally compact, Hausdorff groupoid with a fixed left Haar system. Let G/G denote the orbit space of G and C∗(G) denote the groupoid C∗-algebra. Suppose that the isotropy groups of G are amenable. We show that C∗(G) is CCR if and only if G/G is a T1 topological space and all of the isotropy groups are CCR. We also show that C∗(G) is GCR if and only if G/G is a T0 topological space and all of the isotropy groups are GCR. 1. Intorduction In this paper we generalize two major classification theorems for transformation group C-algebras to theorems that classify groupoid C-algebras. We will denote a transformation group by C(H,X) where H is a locally compact group acting continuously on the left of the locally compact space X . Specifically, in [6], Elliot Gootman showed the following: Theorem 1.1. Suppose H and X are both second countable. Then C(H,X) is GCR if and only if the orbit space is T0 and every stability group is GCR. Dana Williams used a different approach and, in addition to the GCR case, considered the case for CCR transformation group C-algebras. In [17] he proved the theorem below. Theorem 1.2. Suppose that H and X are both second countable. Suppose also that at every point of discontinuity y of the map x 7→ Sx, the stability group Sy is amenable, then C(H,X) is CCR if and only if the orbit space is T1 and the stability groups are CCR. We will prove analogous results to Theorem 1.1 and Theorem 1.2 for groupoid C-algebras. Like Williams’ results, we must still assume that the isotropy groups are amenable. SupposeG is a locally compact, Hausdorff, second countable groupoid with a fixed left Haar system and C(G) is the associated groupoid C-algebra. We show that: Theorem 1.3. The groupoid C-algebra C(G) is CCR if and only if the orbit space is T1 and the isotropy groups are CCR. Theorem 1.4. The groupoid C-algebra C(G) is GCR if and only if the orbit space is T0 and the isotropy groups are GCR. Date: February 2, 2008. 1991 Mathematics Subject Classification. 46L05,46L35.