Groupoid Fest 2006

Abstracts of Talks

Principal Groupoid C*-algebras with Bounded Trace - Lisa Clark

We will define the notion of integrability for groupoids. We then will show for principal groupoids that a groupoid G is integrable if and only if the groupoid C*-algebra C*(G) has bounded trace. This is joint work with Astrid an Huef.

Fell Bundles Associated to Groupoid Morphisms - Valentin Deaconu

Given a continuous open morphism pi: G -> H of etale groupoids with amenable kernel, we construct a Fell bundle E over H and prove that its C*-algebra C*_r(E) is isomorphic to C*_r(G). This is related to results of Fell concerning C*-algebraic bundles over groups. The case H=X, a locally compact space, was treated by Ramazan. We conclude that C*_r(G) is strongly Morita equivalent to a crossed product, the C*-algebra of a Fell bundle arising from an action of the groupoid H on a C*-bundle over H^0. We apply the theory to groupoid morphisms obtained from extensions of dynamical systems and from morphisms of directed graphs with the path lifting property.

Inducing Representations from the Isotropy Group Bundle - Marius Ionescu

One of the most celebrated results in the theory of crossed products of C*-algebras is the proof due to Gootman and Rosenberg based on previous work by Sauvageot of the famous Effros-Hahn conjecture. Its main application is the study of the ideal structure of crossed product C*-algebras. Renault has provided a generalization of the Gootman-Rosenberg proof in the context of groupoid crossed products. In a joint project with Dana Williams, we are looking for conditions on a groupoid dynamical system which imply that all primitive ideals of the groupoid crossed product are induced from a stability group. Such a result would sharpen Renault's results, and more closely parallel the Gootman-Rosenberg-Sauvageot results for ordinary crossed products. In this talk we are going to present some partial results.

Abelian Fell bundles - Alex Kumjian

A Fell bundle over a groupoid H is said to be abelian if E_x is an abelian C*-algebra for all x in H^0. We give some natural examples of abelian Fell bundles and discuss the following result. If E is a saturated abelian Fell bundle over an etale groupoid H, then there is a groupoid G, a covering pi: G -> H and a one-dimensional Fell bundle L over G such that C*_r(L) is isomorphic to C*_r(E).

Amenability for Inverse Semigroups - David Milan

We argue that weak containment is the right notion of amenability for inverse semigroups. Given an inverse semigroup S and a homomorphism phi of S onto a group G, we show S has weak containment if and only if G is amenable and the kernel of phi has weak containment. Using Fell bundle amenability, we extend this result in a way that is suited for inverse semigroups with zero. We show that all graph inverse semigroups have weak containment and that Nica's inverse semigroup of a quasi-lattice ordered group (G,P) has weak containment if and only if (G,P) is amenable in Nica's sense.

Groupoid Methods in Fractal and Wavelet Analysis - Paul Muhly

In this talk, I will discuss work in progress with Marius Ionescu that seeks to place certain constructions that arise in fractal analysis and in wavelet analysis into the setting of groupoids. Our hope is that using groupoids will "explain" certain results in these fields and provide a base from which to attack other problems in the subjects.

A Definition of ``Groupoid Action'' on a C*-Algebra - Alan Paterson

An action of a locally compact group G on a C*-algebra A is just a homomorphism alpha from G into Aut(A) for which the map which sends g to alpha_g(a) is continuous for each a in A. In the case where G is a groupoid with unit space X, the usual definition (Le Gall) is more involved: A has to be a C_0(X)-algebra, and an action is then a C_0(G)-morphism alpha:s^*A -> r^*A such that the map on the C*-bundle A^# over X which sends g to alpha_g is a groupoid homomorphism. For a number of purposes, in particular for groupoid crossed products and the descent homomorphism of groupoid equivariant E-theory, using alpha is awkward, and one would prefer to have a condition involving the map which sends g to alpha_g just as one has in the group case. We show how this can be done. The result is very simple to state: for an action, one just requires there to exist a groupoid homomorphism alpha from G into Aut(A^#) for which the map which sends g to alpha_g(a_s(g)) is continuous for each a in A.

The C*-Envelope of a Semicrossed Product and Nest Representations - Justin Peters

Let X be compact Hausdorff, and phi: X to X a continuous surjection. Let A be the semicrossed product algebra corresponding to the relation f U = U f circ phi. Then the C*-envelope of A is the crossed product of a commutative C*-algebra which contains C(X) as a sub-algebra, with respect to a homeomorphism which we construct. We also show there are "sufficiently many" nest representations. We look at these questions from the groupoid perspective.

Groupoids for Group Representations - Arlan Ramsay

The goal in this talk will be to present some ways in which using groupoids can simplify some aspects of the theory of unitary representations of locally compact groups. Among the topics will be induced representations, group extensions and universal G-spaces.

An Effective Proper Etale Lie Groupoid is Determined, up to Morita Equivalence, by its Quotient Diffeology - Masrour Zoghi

In 1956, Satake introduced the notion of V-manifolds (which later came to be known as orbifolds). He didn't have a good definition of morphisms; with his (somewhat subtle) definition it was not even easy to establish if different data determine equivalent V-manifolds. Later, Haefliger, Moerdijk and others defined orbifolds as proper etale Lie groupoids. For these, good notions of equivalence and morphisms do exist, but they are rather cumbersome to deal with. These notions are what are sometimes called Morita equivalence and Hilsum-Skandalis maps. On the other hand, in 1980, Souriau introduced diffeological spaces, which seem to facilitate the discussion of orbifolds in many instances. What is going to be discussed in this talk is the equivalence of the description of orbifolds as groupoids (in the effective case) and as diffeological spaces.