Furstenberg Transformations and Approximate Conjugacy

Minimal Homeomorphisms and Approximate Conjugacy in Measure

Let X be an infinite compact metric space with finite covering dimension. Let $\afhpa,\bt: X\to X$ be two minimal homeomorphisms. Suppose that the range of $K_0$-groups of both crossed product C*-algebras s are dense in the space of real affine continuous functions. We show that $\af$ and $\bt$ are approximately conjugate uniformly in measure if and only if they have affine homeomorphic invariant probability measure spaces.

Minimal Dynamical Systems and Approximate Conjugacy

Several versions of approximate conjugacy for minimal dynamical systems are introduced. Relation between approximate conjugacy and corresponding crossed product $C^*$-algebras is discussed. For the Cantor minimal systems, a complete description is given for these relations via $K$-theory and $C^*$-algebras. For example, it is shown that two Cantor minimal systems are approximately $τ$-conjugate if and only if they are orbit equivalent and have the same periodic spectrum. It is also shown that two such systems are approximately $K$-conjugate if and only if the corresponding crossed product $C^*…

复分析与算子代数中Furstenberg的$\times 2 \times 3$猜想的表述形式

Formulations of Furstenberg's $\times 2 \times 3$ conjecture in complex analysis and operator algebras

该研究针对遍历论核心开放问题Furstenberg的$\times 2 \times 3$猜想及两个相关命题，分别在复分析和算子代数框架下给出等价表述，形成九组三联猜想，明确了猜想在不同数学领域的对应形式。

On a categorical framework for classifying C*-dynamics up to cocycle conjugacy

We provide the rigorous foundations for a categorical approach to the classification of C*-dynamics up to cocycle conjugacy. Given a locally compact group $G$, we consider a category of (twisted) $G$-C*-algebras, where morphisms between two objects are allowed to be equivariant maps or exterior equivalences, which leads to the concept of so-called cocycle morphisms. An isomorphism in this category is precisely a cocycle conjugacy in the known sense. We show that this category allows sequential inductive limits, and that some known functors on the usual category of $G$-C*-algebras extend. After…

Kishimoto's Conjugacy Theorems in simple $C^*$-algebras of tracial rank one

Let $A$ be a unital separable simple amenable $C^*$-algebra with finite tracial rank which satisfies the Universal Coefficient Theorem (UCT). Suppose $\af$ and $\bt$ are two automorphisms with the Rokhlin property that {induce the same action on the $K$-theoretical data of $A$.} We show that $\af$ and $\bt$ are strongly cocycle conjugate and uniformly approximately conjugate, that is, there exists a sequence of unitaries $\{u_n\}\subset A$ and a sequence of strongly asymptotically inner automorphisms $σ_n$ such that $$ \af={\rm Ad}\, u_n\circ σ_n\circ \bt\circ σ_n^{-1}\andeqn \lim_{n\to\infty}…

有限阿贝尔群作用在Razak-Jacelon代数上的近似可表示性

Approximate representability of finite abelian group actions on the Razak-Jacelon algebra

该研究刻画了有限阿贝尔群外作用与Razak-Jacelon代数恒等映射的张量积作用近似可表示的充要条件，依共轭和上循环共轭分类了此类作用，并构造了模型作用。