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

TL;DR

This paper demonstrates that graph algebras, Exel-Laca algebras, and ultragraph algebras are equivalent up to Morita equivalence, resolving a long-standing open question in the field.

Contribution

It proves the equivalence of these algebra classes up to Morita equivalence and constructs explicit graph models for ultragraph algebras.

Findings

All three classes are Morita equivalent.

Every Exel-Laca algebra is Morita equivalent to a graph algebra.

Characterization of real rank zero for ultragraph algebras.

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.