arXiv subjects

Hung-Chang Liao

Publications and source records attributed to Hung-Chang Liao.

10 recordsLinked to original sources

arXiv2025

Uniform property $Γ$ and the small boundary property

We prove that, for a free action $α\colon G \curvearrowright X$ of a countably infinite discrete amenable group on a compact metric space, the small boundary property is implied by uniform property $Γ$ of the Cartan subalgebra $(C(X) \subseteq C(X) \rtimes_αG)$. The reverse implication has been demonstrated by Kerr and Szabó for free actions, from which we obtain that these two conditions are equivalent. We moreover show that, if $α$ is also minimal, then almost finiteness of $α$ is implied by tracial $\mathcal{Z}$-stability of the subalgebra $(C(X) \subseteq C(X) \rtimes_αG)$. The reverse implication is due to Kerr, resulting in the equivalence of these two properties as well. As an application, we prove that if $α\colon G \curvearrowright X$ and $β\colon H \curvearrowright Y$ are free actions and $α$ has the small boundary property, then $α\times β\colon G \times H \curvearrowright X \times Y$ has the small boundary property. An analogous permanence property is obtained for almost finiteness in case $α$ and $β$ are free minimal actions.

math.OA↗

arXiv2025

Etale equivalence relations with certain prescribed torsion in their homology

Given a non-cyclic simple dimension group D and a subgroup E of Q/Z, we produce a minimal étale equivalence relation R such that H_0(\R) is isomorphic to D \oplus E, where H_0(R) denotes the zeroth homology group of R. The equivalence relation R arises by combining tail-equivalence on a Bratteli diagram with a partial homeomorphism.

math.DS↗

arXiv2023

The diagonal dimension of sub-C*-algebras

We introduce diagonal dimension, a version of nuclear dimension for diagonal sub-C*-algebras (sometimes also referred to as diagonal C*-pairs). Our concept has good permanence properties and detects more refined information than nuclear dimension. In many situations it is precisely how dynamical information is encoded in an associated C*-pair. For free actions on compact Hausdorff spaces, diagonal dimension of the crossed product with its canonical diagonal is bounded above by a product involving Kerr's tower dimension of the action and covering dimension of the space. It is bounded below by the dimension of the space, by the asymptotic dimension of the group, and by the fine tower dimension of the action. For a locally compact, Hausdorff, étale groupoid, diagonal dimension of the groupoid C*-algebra is bounded below by the dynamic asymptotic dimension of the groupoid. For free Cantor dynamical systems, diagonal dimension (defined at the level of the crossed product C*-algebra) and tower dimension (an entirely dynamical notion) agree on the nose. Similarly, for a finitely generated group diagonal dimension of its uniform Roe algebra with the canonical diagonal agrees precisely with asymptotic dimension of the group. This statement also holds for uniformly bounded metric spaces. We apply the lower bounds above to a number of further examples which show how diagonal dimension keeps track of information not seen by nuclear dimension.

math.OA↗

arXiv2021

Almost finiteness, comparison, and tracial $\mathcal{Z}$-stability

Inspired by Kerr's work on topological dynamics, we define tracial $\mathcal{Z}$-stability for sub-$C^*$-algebras. We prove that for a countable discrete amenable group $G$ acting freely and minimally on a compact metrizable space $X$, tracial $\mathcal{Z}$-stability for the sub-$C^*$-algebra $(C(X)\subseteq C(X)\rtimes G)$ implies that the action has dynamical comparison. Consequently, tracial $\mathcal{Z}$-stability is equivalent to almost finiteness of the action, provided that the action has the small boundary property.

math.OA↗

arXiv2020

Comparison and Simplicity of Commutator Subgroups of Full Groups

We show that for a minimal, second countable, locally compact Hausdorff étale groupoid whose unit space is homeomorphic to the Cantor set, if the groupoid has comparison then the commutator subgroup of its full group is simple. This generalizes a result of Bezuglyi and Medynets for Cantor minimal systems and complements Matui's results for topological full groups.

math.DS↗

arXiv2019

A note on crossed products of rotation algebras

We compute the $K$-theory of crossed products of rotation algebras $\mathcal{A}_θ$, for any real angle $θ$, by matrices in $\mathrm{SL}(2,\mathbb{Z})$ with infinite order. Using techniques of continuous fields, we show that the canonical inclusion of $\mathcal{A}_θ$ into the crossed products is injective at the level of $K_0$-groups. We then give an explicit set of generators for the $K_0$-groups and compute the tracial ranges concretely.

math.OA↗

arXiv2017

Isomorphism and Morita equivalence classes for crossed products of irrational rotation algebras by cyclic subgroups of $SL_2(\mathbb{Z})$

Let $θ, θ'$ be irrational numbers and $A, B$ be matrices in $SL_2(\mathbb{Z})$ of infinite order. We compute the $K$-theory of the crossed product $\mathcal{A}_θ\rtimes_A \mathbb{Z}$ and show that $\mathcal{A}_θ \rtimes_A\mathbb{Z}$ and $\mathcal{A}_{θ'} \rtimes_B \mathbb{Z}$ are $*$-isomorphic if and only if $θ= \pmθ' \pmod{\mathbb{Z}}$ and $I-A^{-1}$ is matrix equivalent to $I-B^{-1}$. Combining this result and an explicit construction of equivariant bimodules, we show that $\mathcal{A}_θ \rtimes_A\mathbb{Z}$ and $\mathcal{A}_{θ'} \rtimes_B \mathbb{Z}$ are Morita equivalent if and only if $θ$ and $θ'$ are in the same $GL_2(\mathbb{Z})$ orbit and $I-A^{-1}$ is matrix equivalent to $I-B^{-1}$. Finally, we determine the Morita equivalence class of $\mathcal{A}_θ \rtimes F$ for any finite subgroup $F$ of $SL_2(\mathbb{Z})$.

math.OA↗

arXiv2017

Rokhlin dimension of Z^m actions on simple C*-algebras

We study Rokhlin dimension of Z^m-actions on simple separable stably finite nuclear C*-algebras. We prove that under suitable assumptions, a strongly outer Z^m-action has finite Rokhlin dimension. This extends the known result for automorphisms. As an application, we show that for a large class of C*-algebras, the Z^m-Bernoulli action on the infinite tensor product has finite Rokhlin dimension.

math.OA↗

arXiv2017

Classification of Uniform Roe algebras of locally finite groups

We study the uniform Roe algebras associated to locally finite groups. We show that for two countable locally finite groups $Γ$ and $Λ$, the associated uniform Roe algebras $C^*_u(Γ)$ and $C^*_u(Λ)$ are $*$-isomorphic if and only if their $K_0$ groups are isomorphic as ordered abelian groups with units. This can be seen as a non-separable non-simple analogue of the Glimm-Elliott classification of UHF algebras. To the best of our knowledge, this is the first classification result for a class of non-separable unital $C^*$-algebras. Along the way we also obtain a rigidity result: two countable locally finite groups are bijectively coarsely equivalent if and only if the associated uniform Roe algebras are $*$-isomorphic. Finally, we give a summary of $C^*$-algebraic characterizations for (not necessarily countable) locally finite discrete groups in terms of their uniform Roe algebras. In particular, we show that a discrete group $Γ$ is locally finite if and only if the associated uniform Roe algebra $\ell^\infty(Γ)\rtimes_r Γ$ is locally finite-dimensional.

math.OA↗

arXiv2016

A Rokhlin type theorem for simple C*-algebras of finite nuclear dimension

We study Z-actions on unital simple separable stably finite C*-algebras of finite nuclear dimension. Assuming that the extreme boundary of the trace space is compact and finite dimensional, and that the induced action on the trace space is trivial, we show that strongly outer Z-actions have finite Rokhlin dimension in the sense of Hirshberg, Winter and Zacharias.

math.OA↗