arXiv subjects

Ja A Jeong

Publications and source records attributed to Ja A Jeong.

12 recordsLinked to original sources

arXiv2025

$\mathcal{Z}$-stability for $\mathrm C^*$-algebras of minimal line-bundle-twisted homeomorphisms with the small boundary property

In this paper we show that the Cuntz--Pimsner algebras associated to minimal homeomorphisms twisted by line bundles, along with their orbit-breaking subalgebras, are $\mathcal{Z}$-stable whenever the underlying dynamical system has the small boundary property. This entails that this class is classified by the Elliott invariant. Furthermore, we show that the tensor product of two such $\mathrm{C}^*$-algebras is always $\mathcal{Z}$-stable, without assuming the small boundary property. In particular this applies to $\mathrm{C}^*$-algebras arising from systems with positive mean dimension.

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arXiv2025

Recursive subhomogeneity of orbit-breaking subalgebras of $\mathrm{C}^*$-algebras associated to minimal homeomorphisms twisted by line bundles

In this paper, we construct a recursive subhomogeneous decomposition for the Cuntz--Pimsner algebras obtained from breaking the orbit of a minimal Hilbert $C(X)$-bimodule at a subset $Y \subset X$ with non-empty interior. This generalizes the known recursive subhomogeneous decomposition for orbit-breaking subalgebras of crossed products by minimal homeomorphisms.

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arXiv2023

$\mathrm{C}^*$-algebras associated to homeomorphisms twisted by vector bundles over finite dimensional spaces

In this paper we study Cuntz--Pimsner algebras associated to $\mathrm{C}^*$-correspondences over commutative $\mathrm{C}^*$-algebras from the point of view of the $\mathrm{C}^*$-algebra classification programme. We show that when the correspondence comes from an aperiodic homeomorphism of a finite-dimensional infinite compact metric space $X$ twisted by a vector bundle, the resulting Cuntz--Pimsner algebras have finite nuclear dimension. When the homeomorphism is minimal, this entails classification of these $\mathrm{C}^*$-algebras by the Elliott invariant. This establishes a dichotomy: when the vector bundle has rank one, the Cuntz--Pimsner algebra has stable rank one. Otherwise, it is purely infinite. For a Cuntz--Pimsner algebra of a minimal homeomorphism of an infinite compact metric space $X$ twisted by a line bundle over $X$, we introduce orbit-breaking subalgebras. With no assumptions on the dimension of $X$, we show that they are centrally large subalgebras and hence simple and stably finite. When the dimension of $X$ is finite, they are furthermore $\mathcal{Z}$-stable and hence classified by the Elliott invariant.

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arXiv2018

AF-embeddable labeled graph $C^*$-algebras

Finiteness conditions for $C^*$-algebras like AF-embeddability, quasidiagonality, stable finiteness have been studied by many authors and shown to be equivalent for certain classes of $C^*$-algebras. For example, Schfhauser proves that these conditions are all equivalent for $C^*$-algebras of compact topological graphs, and similar results were established by Clark, an Huef, and Sims for $k$-graph algebras. If $C^*(E,\mathcal L)$ is a labeled graph $C^*$-algebra over finite alphabet, it can be viewed as a $C^*$-algebra of a compact topological graph. For these labeled graph $C^*$-algebras, we provide conditions on labeled paths and show that they are equivalent to AF-embeddability of $C^*(E,\mathcal L)$.

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arXiv2017

Simple labeled graph $C^*$-algebras are associated to disagreeable labeled spaces

By a labeled graph $C^*$-algebra we mean a $C^*$-algebra associated to a labeled space $(E,\mathcal L,\mathcal E)$ consisting of a labeled graph $(E,\mathcal L)$ and the smallest normal accommodating set $\mathcal E$ of vertex subsets. Every graph $C^*$-algebra $C^*(E)$ is a labeled graph $C^*$-algebra and it is well known that $C^*(E)$ is simple if and only if the graph $E$ is cofinal and satisfies Condition (L). Bates and Pask extend these conditions of graphs $E$ to labeled spaces, and show that if a set-finite and receiver set-finite labeled space $(E,\mathcal L, \mathcal E)$ is cofinal and disagreeable, then its $C^*$-algebra $C^*(E,\mathcal L, \mathcal E)$ is simple. In this paper, we show that the converse is also true.

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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).

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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.

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arXiv2014

Finite groups acting on higher dimensional noncommutative tori

For the canonical action $α$ of $\operatorname{SL}_2(\mathbb{Z})$ on 2-dimensional simple rotation algebras $\mathcal{A}_θ$, it is known that if $F$ is a finite subgroup of $\operatorname{SL}_2(\mathbb{Z})$, the crossed products $\mathcal{A}_θ\rtimes_αF$ are all AF algebras. In this paper we show that this is not the case for higher dimensional noncommutative tori. More precisely, we show that for each $n\geq 3$ there exist noncommutative simple $ϕ(n)$-dimensional tori $\mathcal{A}_Θ$ which admit canonical action of $\mathbb{Z}_n$ and for each odd $n\geq 7$ with $2ϕ(n)\geq n+5$ their crossed products $\mathcal{A}_Θ\rtimes_α\mathbb{Z}_n$ are not AF (with nonzero $K_1$-groups). It is also shown that the only possible canonical action by a finite group on a $3$-dimensional simple torus is the flip action by $\mathbb{Z}_2$. Besides, we discuss the canonical actions by finite groups $\mathbb{Z}_5, \mathbb{Z}_8, \mathbb{Z}_{10}$, and $\mathbb{Z}_{12}$ on the $4$-dimensional torus of the form $\mathcal{A}_θ\otimes \mathcal{A}_θ$.

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arXiv2011

The structure of gauge-invariant ideals of labelled graph $C^*$-algebras

In this paper, we consider the gauge-invariant ideal structure of a $C^*$-algebra $C^*(E,\mathcal{L},\mathcal{B})$ associated to a set-finite, receiver set-finite and weakly left-resolving labelled space $(E,\mathcal{L},\mathcal{B})$, where $\mathcal{L}$ is a labelling map assigning an alphabet to each edge of the directed graph $E$ with no sinks. Under the assumption that an accommodating set $\mathcal{B}$ is closed under taking relative complement, it is obtained that there is a one to one correspondence between the set of all hereditary saturated subsets of $\mathcal{B}$ and the gauge-invariant ideals of $C^*(E,\mathcal{L},\mathcal{B})$. For this, we introduce a quotient labelled space $(E,\mathcal{L},[\mathcal{B}]_R)$ arising from an equivalence relation $\sim_R$ on $\mathcal{B}$ and show the existence of the $C^*$-algebra $C^*(E,\mathcal{L},[\mathcal{B}]_R)$ generated by a universal representation of $(E,\mathcal{L},[\mathcal{B}]_R)$. Also the gauge-invariant uniqueness theorem for $C^*(E,\mathcal{L},[\mathcal{B}]_R)$ is obtained. For simple labelled graph $C^*$-algebras $C^*(E,\mathcal{L},\bar{\mathcal{E}})$, where $\bar{\mathcal{E}}$ is the smallest accommodating set containing all the generalized vertices, it is observed that if for each vertex $v$ of $E$, a generalized vertex $[v]_l$ is finite for some $l$, then $C^*(E,\mathcal{L},\bar{\mathcal{E}})$ is simple if and only if $(E,\mathcal{L},\bar{\mathcal{E}})$ is strongly cofinal and disagreeable. This is done by examining the merged labelled graph $(F,\mathcal{L}_F)$ of $(E,\mathcal{L})$ and the common properties that $C^*(E,\mathcal{L},\bar{\mathcal{E}})$ and $C^*(F,\mathcal{L},\bar{\mathcal{F}})$ share.

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arXiv2011

On simple labelled graph $C^*$-algebras

We consider the simplicity of the $C^*$-algebra associated to a labelled space $(E,\CL,\bE)$, where $(E,\CL)$ is a labelled graph and $\bE$ is the smallest accommodating set containing all generalized vertices. We prove that if $C^*(E, \CL, \bE)$ is simple, then $(E, \CL, \bE)$ is strongly cofinal, and if, in addition, $\{v\}\in \bE$ for every vertex $v$, then $(E, \CL, \bE)$ is disagreeable. It is observed that $C^*(E, \CL, \bE)$ is simple whenever $(E, \CL, \bE)$ is strongly cofinal and disagreeable, which is recently known for the $C^*$-algebra $C^*(E, \CL, \CEa)$ associated to a labelled space $(E, \CL, \CEa)$ of the smallest accommodating set $\CEa$.

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arXiv2007

Cancellation for inclusions of C*-algebras of finite depth

Let B be a unital C*-algebra, let A be a unital subalgebra, and let E be a conditional expectation from B to A with index-finite type and a quasi-basis of n elements. Then the topological stable rank satisfies \tsr (B) \leq \tsr (A) + n - 1. As an application, we show that if a unital inclusion A \subset B of C*-algebras has index-finite type and finite depth, and A is simple with stable rank one and Property (SP), then B has cancellation. In particular, if A is a simple unital C*-algebra with stable rank one and Property (SP), and a finite group G acts on A, then the crossed product has cancellation. Separately, if the group is the integers, we obtain cancellation under the additional hypotheses that the group action is outer and is trivial on K_0 (A).

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arXiv2004

Topological entropy and AF subalgebras of graph C*-algebras

Let A_E be the canonical AF subalgebra of a graph C*-algebra C*(E) associated with a locally finite directed graph E. For Brown-Voiculescu's topological entropy ht(Φ_E) of the canonical completely positive map Φ_E on C*(E), ht(Φ_E)=ht(Φ_E|_{A_E})=h_l(E)=h_b(E) is known to hold for a finite graph E, where h_l(E) is the loop entropy of Gurevic and h_b(E) is the block entropy of Salama. For an irreducible infinite graph E, the inequality h_l(E)\leq ht(Φ_E|_{A_E}) has been known recently. It is shown in this paper that ht(Φ_E|_{A_E})\leq max{h_b(E), h_b(tE)}, where tE is the graph E with the direction of the edges reversed. Some irreducible infinite graphs E_p(p>1) with ht(Φ_E|_{A_{E_p}})=log p are also examined.

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