C⁎-algebras associated to Boolean dynamical systems

Abstract

Abstract. The goal of these notes is to present the C

Diagonal-preserving graded isomorphisms of Steinberg algebras

“…• The pair (G E , c), where G E is the groupoid of a directed graph E (see for example [5,7,8,11,13,21]) and c is any continuous cocycle from G E to Γ (see the proof of Theorem 4.1); • the pair (G Λ , c), where G Λ is the groupoid of a finitely-aligned higher-rank graph Λ (see for example [20,36]) and c is any continuous cocycle from G Λ to Γ; • the pair (G tight (T ), c), where G tight (T ) is the tight groupoid constructed from a Boolean dynamical system in [10], and c is any continuous cocycle from…”

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

“…, which is not possible by (8). Hence r(β r ) / ∈ H A 0 for all r ≥ 1, which then easily implies that r(β r ) / ∈H A 0 for all r ≥ 1.…”

“…In [8], the notion of cycle was introduced to define Condition (L B ) for a labeled space (E, L, B) (more generally for Boolean dynamical systems) which can be regarded as another condition analogous to Condition (L) for usual directed graphs. Clearly every cycle is a loop, and if (α, A) is a cycle with an exit, then the exit must be of type (I).…”

“…In the following Theorem 3.7, (a) ⇔ (c) is known in [8,Theorem 9.16] for the Boolean dynamical system induced from a labeled space with the domain property (1). Theorem 3.7.…”

Groupoid Models for the C*-Algebra of Labelled Spaces

“…The main goal of this paper is to prove that the C*-algebra associated with a labelled space is also isomorphic to a groupoid C*-algebra. This was partially done for the countable case in a paper by Carlsen, Ortega and Pardo [8,Theorem 8.3 and Example 11.1] where they work in the setting of Boolean dynamical systems, under an additional assumption on the labelled space. However, certain C*-algebras, such as those associated with shift spaces, when seen from the point of view of labelled graphs, deal with uncountable sets (in the case of the shift, the vertices are the points of the shift).…”