Graph Algebras : Bridges between graph C ∗ - algebras and Leavitt path algebras April 21 – April 26 , 2013 MEALS

Abstract

(in alphabetical order by speaker surname) Speaker: Gene Abrams (University of Colorado at Colorado Springs) Title: Primitive graph algebras Abstract: Let E = (E0, E1, s, r) be an arbitrary directed graph (i.e., no restriction is placed on the cardinality of E0, or of E1, or of s−1(v) for v ∈ E0). Let LK(E) denote the Leavitt path algebra of E with coefficients in a field K, and let C∗(E) denote the graph C∗-algebra of E. (Note: here C∗(E) need not be separable.) We give necessary and sufficient conditions on E so that LK(E) is primitive (joint work with Jason Bell and K.M. Rangaswamy). We then show that these same conditions are precisely the necessary and sufficient conditions on E so that C∗(E) is primitive (joint work with Mark Tomforde). This gives yet another example in a long and growing list of algebraic/analytic properties of the graph algebras LK(E) and C∗(E) for which the graph conditions equivalent to said property are identical, but for which the proof/techniques used are significantly different. In the Leavitt path algebra setting, we show how this result allows for the easy construction of a large collection of prime, non-primitive von Neumann regular algebras (thereby giving a systematic answer to a decades-old question of Kaplansky). In the graph C∗-algebra setting, we show how this result allows for the easy construction of a large collection of prime, non-primitive C∗-algebras (thereby giving a systematic answer to a decades-old question of Dixmier). Speaker: Pere Ara (Universitat Autonoma de Barcelona) Title: Leavitt path algebras of separated graphs and paradoxical decompositions Abstract: A separated graph is a pair (E,C), where E is a directed graph, C = ⊔ v∈E0 Cv, and Cv is a partition of r−1(v) (into pairwise disjoint nonempty subsets) for every vertex v. (In case v is a source, we take Cv to be the empty family of subsets of r−1(v).) Leavitt path algebras LK(E,C) of separated graphs have been recently defined by Goodearl and the presenter [1]. They allow to incorporate the Leavitt algebras of any type (m,n) into the theory of graph algebras. Another method to obtain the Leavitt algebras of type (m,n) has been developed by Hazrat in [4]. Thanks to seminal work of George Bergman, it is possible to explicitly compute the monoids V(LK(E,C)) of finitely generated projective modules over these algebras. In [3], we attach to a finite bipartite separated graph (E,C) a partial dynamical system (Ω(E,C),F, α) possessing a certain universal property. Here Ω(E,C) is a 0-dimensional metrizable compact space, F is a finitely generated free group, and α is a partial action of F on Ω(E,C). The corresponding crossed product algebra CK(Ω(E,C)) oα∗ F is a certain quotient of LK(E,C), and we are able to compute its V-monoid by utilizing a suitable representation as a direct limit of Leavitt path algebras of separated graphs (and the result mentioned above). We will give an application of the theory above to a problem on paradoxical decompositions. Let G be a group acting on a set X. Then a subset E of X is said to be G-paradoxical if E contains two disjoint

Cite this paper

@inproceedings{Tomforde2013GraphA, title={Graph Algebras : Bridges between graph C ∗ - algebras and Leavitt path algebras April 21 – April 26 , 2013 MEALS}, author={Mark Tomforde and Enrique Pardo and Mike Boyle and Toke M. Carlsen and Francesc Perera and E. Jespers M. Ruiz}, year={2013} }