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Abstract Consider a minimal-free topological dynamical system $(X, \mathbb Z^d)$ . It is shown that the radius of comparison of the crossed product C*-algebra $\mathrm {C}(X) \rtimes \mathbb Z^d$ is at most half the mean topological dimension of $(X, \mathbb Z^d)$ . As a consequence, the C*-algebra $\mathrm {C}(X) \rtimes \mathbb Z^d$ is classified by the Elliott invariant if the mean dimension of $(X, \mathbb Z^d)$ is zero. 

We study homeomorphisms of a Cantor set with$k$($k<+\infty$) minimal invariant closed (but not open) subsets; we also study crossed product C*-algebras associated to these Cantor systems and certain of their orbit-cut sub-C*-algebras. In the case where$k\geq 2$, the crossed product C*-algebra is stably finite, has stable rank 2, and has real rank 0 if in addition$(X,\unicode[STIX]{x1D70E})$is aperiodic. The image of the index map is connected to certain directed graphs arising from the Bratteli–Vershik–Kakutani model of the Cantor system. Using this, it is shown that the ideal of the Bratteli diagram (of the Bratteli–Vershik–Kakutani model) must have at least$k$vertices at each level, and the image of the index map must consist of infinitesimals.

 

The class of simple separable KK-contractible (KK-equivalent to \{0\} ) C*-algebra s which have finite nuclear dimension is shown to be classified by the Elliott invariant. In particular, the class of C*-algebras A\otimes K is classifiable, where A is a simple separable C*-algebra with finite nuclear dimension and is the simple inductive limit of Razak algebras with unique trace, which is bounded