On the small boundary property and \mathcal Z-absorption, II

2025/04/04 by Elliott, George A., Niu, Zhuang
#Dynamical Systems (math.DS) #FOS: Mathematics #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.2504.03611

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=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 ≅ A ⊗ \mathcal Z. (3) A has uniform property Γ. (4) l^∞(A)/J_{2, ω, T(A)} has real rank zero. The same statement also holds for unital simple AH algebras with diagonal maps.

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