We generalize Condition (K) from directed graphs to Boolean dynamical systems and show that a locally finite Boolean dynamical system $(\mathcal{ B},\mathcal{ L},\theta)$ with countable $\mathcal{ B}$ and $\mathcal{ L}$ satisfies Condition (K) if and only if every ideal of its $C^*$-algebra is gauge-invariant, if and only if its $C^*$-algebra has the (weak) ideal property, and if and only if its $C^*$-algebra has topological dimension zero. As a corollary we prove that if the $C^*$-algebra of a locally finite Boolean dynamical system with $\mathcal{ B}$ and $\mathcal{ L}$ are countable either has real rank zero or is purely infinite, then $(\mathcal{ B}, \mathcal{ L}, \theta)$ satisfies Condition (K). We also generalize the notion of maximal tails from directed graph to Boolean dynamical systems and use this to give a complete description of the primitive ideal space of the $C^*$-algebra of a locally finite Boolean dynamical system that satisfies Condition (K) and has countable $\mathcal{ B}$ and $\mathcal{ L}$.
Categories: math.OA