Graph algebras, Exel-Laca algebras, and ultragraph algebras coincide up to Morita equivalence

Abstract

Abstract. We prove that the classes of graph algebras, Exel-Laca algebras, and ultragraph algebras coincide up to Morita equivalence. This result answers the long-standing open question of whether every Exel-Laca algebra is Morita equivalent to a graph algebra. Given an ultragraph G we construct a directed graph E such that C * (G) is isomorphic to a full corner of C * (E). As applications, we characterize real rank zero for ultragraph algebras and describe quotients of ultragraph algebras by gauge-invariant ideals.

Realizations of AF-algebras as graph algebras, Exel–Laca algebras, and ultragraph algebras

“…In fact, one of the early results in the theory of graph C * -algebras shows that if A is any AF-algebra, then there is a rowfinite graph E with no sinks such that C * (E) is Morita equivalent to A [7]. From this fact and the result in [16] mentioned above, our three classes (graph C * -algebras, Exel-Laca algebras, and ultragraph C * -algebras) each contain all AF-algebras up to Morita equivalence.…”

“…Ultragraph C * -algebras. Introductory references include [15,16,22,23]. For a set X, let P(X) denote the collection of all subsets of X.…”

On the ideals of ultragraph Leavitt path algebras

“…Ultragraphs and ultragraph C*-algebras were defined by Mark Tomforde in [40] as a unifying approach to C*-algebras associated with infinite matrices (also known as Exel-Laca algebras) and graph C * -algebras. They have proved to be a key ingredient in the study of Morita equivalence of Exel-Laca and graph C *algebras [27]. Recently, Castro, Gonçalves, Royer, Tasca, Wyk, among others, have established nice connections between ultragraph C * -algebras and the symbolic dynamics of shift spaces over infinite alphabets (see [12], [19], [22] and [39]).…”

Simplicity and Chain Conditions for Ultragraph Leavitt Path Algebras via Partial Skew Group Ring Theory

“…Ultragraphs (a generalization of graphs, where the range map takes values on the power set of the vertices) were defined by Tomforde in [17] as a unifying approach to Exel-Laca and graph C * -algebras. They have proved to be a key ingredient in the study of Morita equivalence of Exel-Laca and graph C * -algebras (see [13]) and their representation theory has been studied in [6]. Very recently, ultragraph C * -algebras were connected with the symbolic dynamics of shift spaces over infinite alphabets (see [9,15]) and ultragraphs were the key object behind a new proposal for the generalization of a shift of finite type to the infinite-alphabet case (see [10] for the definition and [3,11] for further developments of the theory).…”