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

TL;DR

This paper constructs two boundary path groupoids from generalized Boolean dynamical systems, proves their topological isomorphism, and shows their $C^*$-algebras are also isomorphic, linking algebraic and topological structures.

Contribution

It introduces two boundary path groupoids associated with generalized Boolean dynamical systems and establishes their topological and algebraic equivalence.

Findings

The tight spectrum is homeomorphic to the boundary path space.

The two boundary path groupoids are topologically isomorphic.

Their $C^*$-algebras are isomorphic to that of the dynamical system.

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.