Condition (K) for Boolean dynamical systems

TL;DR

This paper extends Condition (K) from directed graphs to Boolean dynamical systems, linking it to properties of their $C^*$-algebras and providing a detailed description of their primitive ideal space.

Contribution

It introduces a generalized Condition (K) for Boolean dynamical systems and relates it to ideal properties and topological features of their $C^*$-algebras, also describing primitive ideals.

Findings

Condition (K) characterizes gauge-invariant ideals and topological dimension zero.

Systems with real rank zero or purely infinite $C^*$-algebras satisfy Condition (K).

Primitive ideal space is fully described for systems satisfying Condition (K).

Abstract

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 $\mathcal{L}$.