Lisa Orloff Clark

The representation theory of C*-algebras associated to groupoids

Let E be a second-countable, locally compact, Hausdorff groupoid equipped with an action of T such that G:=E/T is a principal groupoid with Haar system \lambda. The twisted groupoid C*-algebra C*(E;G,\lambda) is a quotient of the C*-algebra of E obtained by completing the space of T-equivariant functions on E. We show that C*(E;G,\lambda) is postliminal if and only if the orbit space of G is T_0 and that C*(E;G, \lambda) is liminal if and only if the orbit space is T_1. We also show that C*(E;G, \lambda) has bounded trace if and only if G is integrable and that C*(E;G, \lambda) is a Fell algebra if and only if G is Cartan. Let \G be a second-countable, locally compact, Hausdorff groupoid with Haar system \lambda and continuously varying, abelian isotropy groups. Let A be the isotropy groupoid and R := \G/A. Using the results about twisted groupoid C*-algebras, we show that the C*-algebra C*(\G, \lambda) has bounded trace if and only if R is integrable and that C*(\G, \lambda) is a Fell algebra if and only if R is Cartan. We illustrate our theorems with examples of groupoids associated to directed graphs.

Using Steinberg algebras to study decomposability of Leavitt path algebras

Abstract Given an arbitrary graph E we investigate the relationship between E and the groupoid G E {G_{E}} . We show that there is a lattice isomorphism between the lattice of pairs ( H , S ) {(H,S)} , where H is a hereditary and saturated set of vertices and S is a set of breaking vertices associated to H, onto the lattice of open invariant subsets of G E ( 0 ) {G_{E}^{(0)}} . We use this lattice isomorphism to characterise the decomposability of the Leavitt path algebra L K ⁢ ( E ) {L_{K}(E)} , where K is a field. First we find a graph condition to characterise when an open invariant subset of G E ( 0 ) {G_{E}^{(0)}} is closed. Then we give both a graph condition and a groupoid condition each of which is equivalent to L K ⁢ ( E ) {L_{K}(E)} being decomposable in the sense that it can be written as a direct sum of two nonzero ideals. We end by relating decomposability of a Leavitt path algebra with the existence of nontrivial central idempotents. In fact, all the nontrivial central idempotents can be described.

Cohn Path Algebras of Higher-Rank Graphs

In this article, we introduce Cohn path algebras of higher-rank graphs. We prove that for a higher-rank graph Λ, there exists a higher-rank graph T Λ such that the Cohn path algebra of Λ is isomorphic to the Kumjian-Pask algebra of T Λ. We then use this isomorphism and properties of Kumjian-Pask algebras to study Cohn path algebras. This includes proving a uniqueness theorem for Cohn path algebras.

SUBSETS OF VERTICES GIVE MORITA EQUIVALENCES OF LEAVITT PATH ALGEBRAS

We show that every subset of vertices of a directed graph E gives a Morita equivalence between a subalgebra and an ideal of the associated Leavitt path algebra. We use this observation to prove an algebraic version of a theorem of Crisp and Gow: certain subgraphs of E can be contracted to a new graph G such that the Leavitt path algebras of E and G are Morita equivalent. We provide examples to illustrate how desingularising a graph, and in- or out-delaying of a graph, all fit into this setting.