The Generalized Effros-Hahn Conjecture for Groupoids

Ideals of etale groupoid algebras and Exel's Effros-Hahn conjecture

We extend to arbitrary commutative base rings a recent result of Demeneghi that every ideal of an ample groupoid algebra over a field is an intersection of kernels of induced representations from isotropy groups, with a much shorter proof, by using the author's Disintegration Theorem for groupoid representations. We also prove that every primitive ideal is the kernel of an induced representation from an isotropy group; however, we are unable to show, in general, that it is the kernel of an irreducible induced representation. If each isotropy group is finite (e.g., if the groupoid is principal)…

AF $C^*$-algebras from non AF groupoids

We construct ample groupoids from certain categories of paths, and prove that their $C^*$-algebras coincide with the continued fraction AF algebras of Effros and Shen. The proof relies on recent classification results for simple nuclear $C^*$-algebras. The groupoids are not principal. This provides examples of Cartan subalgebras in the continued fraction AF algebras that are isomorphic, but not conjugate, to the standard diagonal subalgebras.

Fredholm criteria for pseudodifferential operators and induced representations of groupoid algebras

We characterize the groupoids for which an operator is Fredholm if, and only if, its principal symbol and all its boundary restrictions are invertible. A groupoid with this property is called {\em Fredholm}. Using results on the Effros-Hahn conjecture, we show that an almost amenable, Hausdorff, second countable groupoid is Fredholm. Many groupoids, and hence many pseudodifferential operators appearing in practice, fit into this framework. In particular, one can use these results to characterize the Fredholm operators on manifolds with cylindrical and poly-cylindrical ends, on manifolds that a…

带有层系数的 étale 群胚代数的理想及其在拓扑动力学中的应用

Ideals of étale groupoid algebras with coefficients in a sheaf with applications to topological dynamics

本文证明了带层系数群胚代数的 Effros-Hahn 猜想，刻画了其理想及代数性质，并将其应用于逆半群作用的拓扑动力学及 C*-代数的单性判别。