C*-algebras of a Cantor system with finitely many minimal subsets: structures, K-theories, and the index map

TL;DR

This paper analyzes the structure and K-theory of C*-algebras arising from Cantor systems with finitely many minimal subsets, revealing their stable properties and the nature of their index maps.

Contribution

It characterizes the C*-algebras associated with Cantor systems with multiple minimal subsets, including their stable finiteness, rank, and the structure of the index map.

Findings

C*-algebras are stably finite with stable rank 2

Real rank zero when the system is aperiodic

Index map image relates to directed graphs from Bratteli-Vershik models

Abstract

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 their certain orbit-cut sub-C*-algebras. In the case that $k\geq 2$, the crossed product C*-algebra is stably finite, has stable rank 2, and has real rank zero if in addition $(X, \sigma)$ 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 infinitesimals.