Simple labeled graph C⁎-algebras are associated to disagreeable labeled spaces

Abstract

By a labeled graph C * -algebra we mean a C * -algebra associated to a labeled space (E, L, E ) consisting of a labeled graph (E, L) and the smallest normal accommodating set E of vertex subsets. Every graph C * -algebra C * (E) is a labeled graph C * -algebra and it is well known that C * (E) is simple if and only if the graph E is cofinal and satisfies Condition (L). Bates and Pask extend these conditions of graphs E to labeled spaces, and show that if a set-finite and receiver set-finite labeled space (E, L, E ) is cofinal and disagreeable, then its C * -algebra C * (E, L, E ) is simple. In this paper, we show that the converse is also true.2010 Mathematics Subject Classification. 46L05, 46L55.

The simplicity of the C*-algebras associated to arbitrary labeled spaces

“…So, we can weaken the assumption as follows. The idea of proof is same with [15,Lemma 3.6]. We have only included the proof of Lemma 4.10(1) for completeness.…”

“…It then is known in [14,Proposition 3.7] that if (E, L, B) is disagreeable, then (E, L, B) satisfies Condition (L). But, the converse is not true, in general ( [15,Proposition 3.2]). Based on these concepts, it is eventually known in [15, Theorem 3.17] that for a set-finite and receiver set-finite labeled space (E, L, E) with E having no sinks or sources, C * (E, L, E) is simple if and only if (E, L, E) is strongly cofinal in the sense of [15, Definition 2.10] and disagreeable.…”

“…Based on these concepts, it is eventually known in [15, Theorem 3.17] that for a set-finite and receiver set-finite labeled space (E, L, E) with E having no sinks or sources, C * (E, L, E) is simple if and only if (E, L, E) is strongly cofinal in the sense of [15, Definition 2.10] and disagreeable. It is also prove in [15,Theorem 3.17] that for a labeled space whose Boolean dynamical system satisfies a sort of domain condition, C * (E, L, E) is simple if and only if (E, L, E) satisfies Condition (L) and there are no nonempty hereditary saturated subsets of E.…”

“…We generalize these results to labeled graph C * -algebras associated to arbitrary labeled spaces. We first give an example that shows why we need to change the definition of strong cofinality given in [10,Definition 3.1] to [15,Definition 2.10]. We next prove that (E, L, E) is strongly cofinal and for any A ∈ E \ {∅} and B ∈ E, there is C ∈ E reg such that B \ C ∈ H(A) if and only if (E, L, E) is minimal, in the sense that {∅} and E are the only hereditary saturated subsets of E. We also identify the minimal condition with the property that an ideal containing one vertex projection must be the whole C * -algebra (Theorem 4.6).…”

“…Remark 4.11. For a set-finite and receiver set-finite labeled space (E, L, E) with E having no sinks or sources, it is shown in[15, Theorem 3.7] that if C * (E, L, E) is simple, then (E, L, E) is disagreeable. Thus, if C * (E, L,E) is simple, the labeled space (E, L, E) can not have a nonempty set A ∈ E and an irreducible path β ∈ L * (E) such that L(AE n|β| ) = {β n } for all n ≥ 1.…”