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Journal of Operator Theory

Volume 90, Issue 2, Autumn 2023  pp. 523-544.

Sturmian subshifts and their $C^*$-algebras

Authors:  Kevin Aguyar Brix
Author institution: School of Mathematics and Statistics, University of Glasgow, Glasgow G12 8QQ, U.K.

Summary:  This paper investigates the structure of $C^*$-algebras built from one-sided Sturmian subshifts. They are parametrised by irrationals in the unit interval and built from a local homeomorphism associated to the subshift. We provide an explicit construction and description of this local homeomorphism. The $C^*$-algebras are $*$-isomorphic exactly when the systems are conjugate, and they are Morita equivalent exactly when the defining irrationals are equivalent (this happens precisely when the systems are flow equivalent). Using only elementary dynamical tools, we compute the dynamic asymptotic dimension of the (groupoid of the) local homeomorphism to be one, and by a result of Guentner, Willett, and Yu, it follows that the nuclear dimension of the $C^*$-algebras is one.

Keywords: Sturmian subshifts, $C^*$-algebras, nuclear dimension

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