Draft of dissertation:
In this paper we study properties of groupoids by looking at their $C^∗$-algebras. We introduce a notion of rapid decay for transformation groupoids and we show that this is equivalent to the underlying group having the property of rapid decay. We show that our definition is equivalent to a number of other properties which are in direct correspondence to the group case. Additionally, given two bilipschitz equivalent discrete groups we construct an isomorphism of the corresponding transformation groupoids and are able to reformulate the open problem of showing invariance of rapid decay under quasi-isometry.
We then begin to examine various notions of amenability when abstracted to measured ́etale groupoids. In the group case, the following properties are equivalent:
1) $G$ is amenable
3) The trivial representation decends from ^C^∗(G)$ to $C_r^∗(G)$.
In the groupoid, $G$, case we have 1) ⇒ 2) ⇒ 3), but it was shown by Rufus Willett that $C_r^∗(G) = C^∗(G)$ is not enough in general to give amenability of G. In this paper we study property 3) for groupoids, formulate some equivalent statements and show that 3) ⇒ 2) is also false in general.
Jul 19 @ 3:00 pm – 4:30 pm
Mar 22 – Mar 24 all-day