Boundary path groupoids of generalized Boolean dynamical systems and their C⁎-algebras

Abstract

In this paper, we provide two types of boundary path groupoids from a generalized Boolean dynamical system (B,L,θ,Iα). For the first groupoid, we associate an inverse semigroup to a generalized Boolean dynamical system and use the tight spectrum T as the unit space of a groupoid Γ(B,L,θ,Iα) that is isomorphic to the tight groupoid Gtight. The other one is defined as the Renault-Deaconu groupoid Γ(∂E,σE) arising from a topological correspondence E associated with a generalized Boolean dynamical system. We then prove that the tight spectrum T is homeomorphic to the boundary path space ∂E obtained from the topological correspondence. Using this result, we prove that the groupoid Γ(B,L,θ,Iα) equipped with the topology induced from the topology on Gtight is isomorphic to Γ(∂E,σE) as a topological groupoid. Finally, we show that their C⁎-algebras are isomorphic to the C⁎-algebra of the generalized Boolean dynamical system.