A Cuntz--Krieger Uniqueness theorem for C*-algebras of relative generalized Boolean dynamical systems

We prove a version of the Cuntz–Krieger Uniqueness Theorem for $C^{\ast }$-algebras of arbitrary relative generalized Boolean dynamical systems. We then describe properties of a $C^{\ast }$-algebra of a relative generalized Boolean dynamical system when the underlying Boolean dynamical system satisfies Condition (K). We also define a notion of minimality of a Boolean dynamical system and give sufficient and necessary conditions for the minimality. Using these results, we characterize the generalized Boolean dynamical systems who’s $C^{\ast }$-algebra is simple.