arXiv subjects

Eun Ji Kang

Publications and source records attributed to Eun Ji Kang.

12 recordsLinked to original sources

arXiv2026

Pullbacks of Groupoid $C^*$-Algebras over Disjoint Invariant Sets

We establish a pullback theorem for $C^*$-algebras of locally compact Hausdorff étale groupoids. The theorem shows that, when the unit space is covered by two closed invariant subsets whose complements are disjoint open invariant subsets, the corresponding groupoid $C^*$-algebras form a pullback diagram in the category of $\mathbb T$-$C^*$-algebras and $\mathbb T$-equivariant $*$-homomorphisms, for the gauge actions induced by a $\mathbb Z$-valued cocycle and its restrictions. We then prove a collection of boundary-path decomposition theorems for graphs, relative graphs, and topological graphs. We show that admissible decompositions of graphs, together with their analogues in the relative and topological settings, induce corresponding decompositions of boundary path spaces. Combining these decomposition theorems with the groupoid pullback theorem, we recover previously known pullback theorems for graph $C^*$-algebras, relative graph $C^*$-algebras, and topological graph $C^*$-algebras. Thus these pullback phenomena are explained by a single groupoid-theoretic mechanism.

math.OA↗

arXiv2025

K-Theory and Structural Properties of $C^*$-Algebras Associated with Relative Generalized Boolean Dynamical Systems

We present an explicit formula for the $K$-theory of the $C^*$-algebra associated with a relative generalized Boolean dynamical system $(\CB, \CL, θ, \CI_\af; \CJ)$. In particular, we find concrete generators for the $K_1$-group of $C^*(\CB, \CL, θ, \CI_\af; \CJ)$. We also prove that every gauge-invariant ideal of $C^*(\CB, \CL, θ, \CI_\af; \CJ)$ is Morita equivalent to a $C^*$-algebra of a relative generalized Boolean dynamical system. As a structural application, we show that if the underlying Boolean dynamical system $(\CB, \CL, θ)$ satisfies Condition (K), then the associated $C^*$-algebra is $K_0$-liftable. Furthermore, we deduce that if $C^*(\CB, \CL, θ, \CI_\af; \CJ)$ is separable and purely infinite, then it has real rank zero.

math.OA↗

arXiv2024

A Categorical Interpretation of Continuous Orbit Equivalence for Partial Dynamical Systems

We define the orbit morphism of partial dynamical systems and prove that an orbit morphism being an isomorphism in the category of partial dynamical systems and orbit morphisms is equivalent to the existence of a continuous orbit equivalence between the given partial dynamical systems that preserves the essential stabilisers. We show that this is equivalent to the existence of a diagonal-preserving isomorphism between the corresponding crossed products when the essential stabilisers of partial actions are torsion-free and abelian. We also characterize when an étale groupoid is isomorphic to the transformation groupoid of some partial action. Additionally, we explore the implications in the context of semi-saturated orthogonal partial dynamical systems over free groups, establishing connections with Deaconu-Renault systems and the concept of eventual conjugacy. Finally, we apply our results to C*-algebras associated with generalized Boolean dynamical systems.

math.OA↗

arXiv2024

Representing topological full groups in Steinberg algebras and C*-algebras

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.

math.OA↗

arXiv2023

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.

math.OA↗

arXiv2023

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 who's $C^*$-algebra is simple.

math.OA↗

arXiv2022

C*-algebras of generalized Boolean dynamical systems as partial crossed products

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.

math.OA↗

arXiv2021

Boundary path groupoids of generalized Boolean dynamical systems and their C*-algebras

In this paper, we provide two types of boundary path groupoids from a generalized Boolean dynamical system $(\mathcal{B},\mathcal{L}, θ, \mathcal{I}_α)$. 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 $Γ(\mathcal{B},\mathcal{L}, θ, \mathcal{I}_α)$ that is isomorphic to the tight groupoid $\mathcal{G}_{tight}$. The other one is defined as the Renault-Deaconu groupoid $Γ(\partial E, σ_E)$ 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 $Γ(\mathcal{B},\mathcal{L}, θ, \mathcal{I}_α)$ equipped with the topology induced from the topology on $\mathcal{G}_{tight}$ is isomorphic to $Γ(\partial E, σ_E)$ as a topological groupoid. Finally, we show that their $C^*$-algebras are isomorphic to the $C^*$-algebra of the generalized Boolean dynamical system.

math.OA↗

arXiv2021

Condition (K) for Boolean dynamical systems

We generalize Condition (K) from directed graphs to Boolean dynamical systems and show that a locally finite Boolean dynamical system $(\mathcal{B},\mathcal{L},θ)$ 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 $(\mathcal{B}, \mathcal{L}, θ)$ 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 $\mathcal{L}$.

math.OA↗

arXiv2021

Gauge-invariant ideals of C*-algebras of Boolean dynamical systems

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.

math.OA↗

arXiv2017

Purely infinite labeled graph $C^*$-algebras

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 Rørdam 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).

math.OA↗

arXiv2016

Finite simple labeled graph $C^*$-algebras of Cantor minimal subshifts

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.

math.OA↗