Mean ergodic theorem for amenable discrete quantum groups and a Wiener type theorem for compact metrizable groups

Construction of minimal skew products of amenable minimal dynamical systems

For an amenable minimal topologically free dynamical system $α$ of a group on a compact metrizable space $Z$ and for a compact metrizable space $Y$ satisfying a mild condition, we construct a minimal skew product extension of $α$ on $Z\times Y$. This generalizes a result of Glasner and Weiss. We also study the pure infiniteness of the crossed products of minimal dynamical systems arising from this result. In particular, we give a generalization of a result of Rørdam and Sierakowski.

可数群作用的非交换Wiener-Wintner定理

Non-Commutative Wiener-Wintner theorem for amenable group actions

本文针对有限von Neumann代数上保迹自同构的可数群作用，建立了Jacobs-de Leeuw-Glicksberg分解，并证明了非交换Wiener-Wintner定理。

可数群作用的比较性与几乎有限性

Comparison and Almost Finiteness for Actions of Amenable Groups

本文证明可数无限离散顺从群在零维或可度量空间上的作用具有动力学比较性，并由此建立几乎有限性，解决多个猜想与开放问题。

变换群代数的单生成性

Single generation of transformation group algebras

研究有限生成离散可和群作用下变换群代数的单生成性，证明满足自由、极小、一致Rokhlin性质及开集Cuntz比较条件时C(X)⋊Γ单生成，且自由极小ℤᵈ交叉积也单生成。

几乎有限性与动力起源的群

Almost finiteness and groups of dynamical origin

本文提出良好子群性质，证明其对顺从群作用蕴含比较性，从而确立多类动力起源群在有限维空间上的自由作用几乎有限，并得到可分类交叉积。

Invariant ergodic measures and the classification of crossed product $C^\ast$-algebras

Let $α: G\curvearrowright X$ be a minimal free continuous action of an infinite countable amenable group on an infinite compact metrizable space. In this paper, under the hypothesis that the invariant ergodic probability Borel measure space $E_G(X)$ is compact and zero-dimensional, we show that the action $α$ has the small boundary property. This partially answers an open problem in dynamical systems that asks whether a minimal free action of an amenable group has the small boundary property if its space $M_G(X)$ of invariant Borel probability measures forms a Bauer simplex. In addition, under…