Representations and the reduction theorem for ultragraph Leavitt path algebras

Abstract

In this paper we study representations of ultragraph Leavitt path algebras via branching systems and, using partial skew ring theory, prove the reduction theorem for these algebras. We apply the reduction theorem to show that ultragraph Leavitt path algebras are semiprime and to completely describe faithfulness of the representations arising from branching systems, in terms of the dynamics of the branching systems. Furthermore, we study permutative representations and provide a sufficient criteria for a permutative representation of an ultragraph Leavitt path algebra to be equivalent to a representation arising from a branching system. We apply this criteria to describe a class of ultragraphs for which every representation (satisfying a mild condition) is permutative and has a restriction that is equivalent to a representation arising from a branching system.

Irreducible and permutative representations of ultragraph Leavitt path algebras

“…Suppose that G satisfies Condition (L). By Theorem 5.1 in [25] ϕ is faithful. Now suppose that G does not satisfy Condition (L).…”

“…Iterated function systems and branching systems are widely used in the study of representations of algebras associated to combinatorial objects, see for example [8,11,12,13,14,15,16,17,18,19,20,21,23,25,31]. Hence it is interesting to note that the representation π of Theorem 3.7 can be constructed via branching systems.…”

“…For example, in this section we will completely characterize when π is faithful, a result for ultragraphs that also improves Proposition 4.4 in [10] regarding Leavitt path algebras. Before we proceed we recall the following relevant definitions (as in [25]). Definition 4.1 Let G be an ultragraph, X be a set and let {R e , D A } e∈G 1 ,A∈G 0 be a family of subsets of X.…”

“…In Proposition 4.5 of [25] it is shown that a branching system induces a representation of L R (G) in End(M(X)). For our purposes, we will consider the following branching system on p * ∪ p ∞ .…”

“…Ultragraphs (and their algebras) are generalizations of graphs (and their algebras) with applications in Symbolic Dynamics, Operator Algebras and Algebra, see for example [9,15,24,25,26,27,28,29,30,33,34,36,39,40]. The Leavitt path algebra associated to an ultragraph was defined in [32],…”