2021/06/17 by de Castro, Gilles G., Kang, Eun Ji · 1 citation
#46L05 #FOS: Mathematics #Operator Algebras (math.OA) #Primary: 46L55 #Secondary: 37B99
paper · doi:10.48550/arxiv.2106.09366
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.