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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., Comment: 33 pages

We study the natural representation of the topological full group of an ample Hausdorff groupoid in the groupoid's complex Steinberg algebra and in its full and reduced C*-algebras. We characterise precisely when this representation is injective and show that it is rarely surjective. We then restrict our attention to discrete groupoids, which provide unexpected insight into the behaviour of the representation of the topological full group in the full and reduced groupoid C*-algebras. We show that the image of the representation is not dense in the full groupoid C*-algebra unless the groupoid is a group, and we provide an example showing that the image of the representation may still be dense in the reduced groupoid C*-algebra even when the groupoid is not a group., Comment: This version matches the version in the Journal of Mathematical Analysis and Applications. 12 pages

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 who's $C^*$-algebra is simple., Comment: 26 pages

In this paper, we realize C*-algebras of generalized Boolean dynamical systems as partial crossed products. Reciprocally, we give some sufficient conditions for a partial crossed product to be isomorphic to a C*-algebra of a generalized Boolean dynamical system. As an application, we show that gauge-invariant ideals of C*-algebras of generalized Boolean dynamical systems are themselves C*-algebras of generalized Boolean dynamical system., Comment: 28 pages. arXiv admin note: text overlap with arXiv:2106.09366

In this paper, we provide two types of boundary path groupoids from a generalized Boolean dynamical system $(\mathcal{B},\mathcal{L}, \theta, \mathcal{I}_{\alpha})$. For the first groupoid, we associate an inverse semigroup to a generalized Boolean dynamical system and use the tight spectrum $\mathsf{T}$ as the unit space of a groupoid $\Gamma(\mathcal{B},\mathcal{L}, \theta, \mathcal{I}_{\alpha})$ that is isomorphic to the tight groupoid $\mathcal{G}_{tight}$. The other one is defined as the Renault-Deaconu groupoid $\Gamma(\partial E, \sigma_E)$ arising from a topological correspondence $E$ associated with a generalized Boolean dynamical system. We then prove that the tight spectrum $\mathsf{T} $ is homeomorphic to the boundary path space $\partial E$ obtained from the topological correspondence. Using this result, we prove that the groupoid $\Gamma(\mathcal{B},\mathcal{L}, \theta, \mathcal{I}_{\alpha})$ equipped with the topology induced from the topology on $\mathcal{G}_{tight}$ is isomorphic to $\Gamma(\partial E, \sigma_E)$ as a topological groupoid. Finally, we show that their $C^*$-algebras are isomorphic to the $C^*$-algebra of the generalized Boolean dynamical system., Comment: 42 pages

We enlarge the class of $C^*$-algebras of Boolean dynamical systems in order to include all weakly left-resolving normal labelled space $C^*$-algebras in it. We prove a gauge-invariant uniqueness theorem and classify all gauge-invariant ideals of these $C^*$-algebras of generalized Boolean dynamical systems and describe the corresponding quotients as $C^*$-algebras of relative generalized Boolean dynamical systems., Comment: 28 pages. Version 2 is a slightly revised version of version 1, and is the version that was published in Journal of Mathematical Analysis and Applications. In version 2 (and the published version), there were some small mistakes on page 24 and in the proof of Lemma 7.2.(2) on page 25. They have been corrected in version 3 (see also https://doi.org/10.1016/j.jmaa.2020.124754)

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 $\mathcal{L}$., Comment: 25 pages. Version 2 is a minor update of version 1 and is the version that will be published in J. Aust. Math. Soc

In this paper, we consider pure infiniteness of generalized Cuntz-Krieger algebras associated to labeled spaces $(E,\mathcal{L},\mathcal{E})$. It is shown that a $C^*$-algebra $C^*(E,\mathcal{L},\mathcal{E})$ is purely infinite in the sense that every nonzero hereditary subalgebra contains an infinite projection (we call this property (IH)) if $(E, \mathcal{L},\mathcal{E})$ is disagreeable and every vertex connects to a loop. We also prove that under the condition analogous to (K) for usual graphs, $C^*(E,\mathcal{L},\mathcal{E})=C^*(p_A, s_a)$ is purely infinite in the sense of Kirchberg and R{\o}rdam if and only if every generating projection $p_A$, $A\in \mathcal{E}$, is properly infinite, and also if and only if every quotient of $C^*(E,\mathcal{L},\mathcal{E})$ has the property (IH)., Comment: 32 pages

It is now well known that a simple graph $C^*$-algebra $C^*(E)$ of a directed graph $E$ is either AF or purely infinite. In this paper, we address the question of whether this is the case for labeled graph $C^*$-algebras recently introduced by Bates and Pask as one of the generalizations of graph $C^*$-algebras, and show that there exists a family of simple unital labeled graph $C^*$-algebras which are neither AF nor purely infinite. Actually these algebras are shown to be isomorphic to crossed products $C(X)\times_T \mathbb Z$ where the dynamical systems $(X,T)$ are Cantor minimal subshifts. Then it is an immediate consequence of well known results about this type of crossed products that each labeled graph $C^*$-algebra in the family obtained here is an $A\mathbb T$ algebra with real rank zero and has $\mathbb Z$ as its $K_1$-group., Comment: 18 pages