Title: On the small boundary property and $\mathcal Z$-absorption, II Show Chinese title

Title: 关于小边界性质和$\mathcal Z$-吸收，II

Abstract: Consider a minimal and free topological dynamical system $(X, \mathbb Z^d)$. It is shown that zero mean dimension of $(X, \mathbb Z^d)$ is characterized by $\mathcal Z$-absorption of the crossed product C*-algebra $A=\mathrm{C}(X) \rtimes \mathbb Z^d$, where $\mathcal Z$ is the Jiang-Su algebra. In fact, among other conditions, the following are shown to be equivalent: (1) $(X, \mathbb Z^d)$ has the small boundary property. (2) $A \cong A \otimes \mathcal Z$. (3) $A$ has uniform property $\Gamma$. (4) $l^\infty(A)/J_{2, \omega, \mathrm{T}(A)}$ has real rank zero. The same statement also holds for unital simple AH algebras with diagonal maps. Show Chinese abstract

Abstract: 考虑一个极小且自由的拓扑动力系统$(X, \mathbb Z^d)$。 证明了$(X, \mathbb Z^d)$的零均维数由交叉积C*-代数$A=\mathrm{C}(X) \rtimes \mathbb Z^d$的$\mathcal Z$吸收性来刻画，其中$\mathcal Z$是 Jiang-Su 代数。 事实上，在其他条件中，以下条件被证明是等价的： (1)$(X, \mathbb Z^d)$具有小边界性质。 (2)$A \cong A \otimes \mathcal Z$。 (3) $A$ 具有一致性质 $\Gamma$。 (4) $l^\infty(A)/J_{2, \omega, \mathrm{T}(A)}$ 的实秩为零。 该陈述对于具有对角映射的单位单AH代数也同样成立。