C∗-ACTIONS OF r-DISCRETE GROUPOIDS AND INVERSE SEMIGROUPS

Many important C∗-algebras, such as AF-algebras, Cuntz-Krieger algebras, graph algebras and foliation C∗-algebras, are the C∗-algebras of r-discrete groupoids. These C∗-algebras are often associated with inverse semigroups through the C∗-algebra of the inverse semigroup [HR90] or through a crossed product construction as in Kumjian’s localization [Kum84]. Nica [Nic94] connects groupoid C∗-algebras with the partial crossed product C∗-algebras of Exel [Exe94] and McClanahan [McC95]. This gives another connection between groupoid C∗-algebras and inverse semigroup C∗algebras since [Sie97b] and [Exear] show that discrete partial crossed products are basically special cases of the inverse semigroup crossed products of [Sie97b], [Patar], [Sie97a]. The heart of these connections is Renault’s observation in [Ren80] that an r-discrete groupoid can be recovered from the way the inverse semigroup of open G-sets acts on the unit space of the groupoid. In the upcoming [Patar] Paterson further develops this connection by showing that the C∗-algebra of an r-discrete groupoid G is the crossed product of C0(G 0) by the action of the inverse semigroup of open G-sets. The purpose of this paper is to explore this connection on the level of C∗-crossed products. Renault defines [Ren87] a C∗-action of a groupoid as a functor to the category of C∗-algebras and homomorphisms, in which the collection of object C∗-algebras are glued together as a C∗-bundle over G0 and the action is appropriately continuous. We associate to this an action of any sufficiently large inverse semigroup S of open G-sets on the C0-section algebra of the bundle. Conversely, starting with an action (satisfying certain mild conditions) of S on a C∗-algebra B, we obtain an associated C∗-bundle over G0 via the realization that C0(G 0) will act as central multipliers of B. Then we construct the groupoid action using the expected “germs of local automorphisms” approach that goes back to [Hae58] and [Rei83]. The C∗-bundles arising this way are typically only upper semicontinuous, rather than continuous. So we use a slight generalization of Renault’s theory. The philosophy is that inverse semigroups and r-discrete groupoids are two sides of the same coin; passing back and forth between groupoid and inverse semigroup constructions may benefit both theories. The theory of groupoid C∗-algebras is more developed, but the inverse semigroup theory is more algebraic. For example, we can show that the C∗-algebra of an r-discrete groupoid is an enveloping C∗-algebra without using Renault’s disintegration theorem. In fact one could suspect that for r-discrete groupoids the disintegration theorem follows from the less complicated inverse semigroup disintegration theorem. Other applications could include inverse semigroup versions