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 $(\mathcal{B},\mathcal{L}, \theta, \mathcal{I}_{\alpha})$. For the first groupoid, we associate an inverse semigroup to a generalized Boolean dynamical system and use the tight spectrum $\mathsf{T}$ as the unit space of a groupoid $\Gamma(\mathcal{B},\mathcal{L}, \theta, \mathcal{I}_{\alpha})$ that is isomorphic to the tight groupoid $\mathcal{G}_{tight}$. The other one is defined as the Renault-Deaconu groupoid $\Gamma(\partial E, \sigma_E)$ arising from a topological correspondence $E$ associated with a generalized Boolean dynamical system. We then prove that the tight spectrum $\mathsf{T} $ is homeomorphic to the boundary path space $\partial E$ obtained from the topological correspondence. Using this result, we prove that the groupoid $\Gamma(\mathcal{B},\mathcal{L}, \theta, \mathcal{I}_{\alpha})$ equipped with the topology induced from the topology on $\mathcal{G}_{tight}$ is isomorphic to $\Gamma(\partial E, \sigma_E)$ as a topological groupoid. Finally, we show that their $C^*$-algebras are isomorphic to the $C^*$-algebra of the generalized Boolean dynamical system.