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∗-algebras of arbitrary relative generalized Boolean dynamical systems. We then describe properties of a C∗-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 whose C*-algebra is simple.