A generalized uniqueness theorem for generalized Boolean dynamical systems

We characterize the canonical diagonal subalgebra of the C*-algebra associated with a generalized Boolean dynamical system. We also introduce a particular commutative subalgebra, which we call the abelian core, in our C*-algebra. We then establish a uniqueness theorem under the assumptions that B and L are countable, which says that a *-homomorphism of our C*-algebra is injective if and only if its restriction to the abelian core is injective.