A note on the nuclear dimension of Cuntz-Pimsner $C^*$-algebras associated with minimal shift spaces

TL;DR

This paper investigates the nuclear dimension of Cuntz-Pimsner $C^*$-algebras associated with minimal shift spaces, showing it equals 1 when the space has finitely many left special elements, using detailed cover analysis.

Contribution

It establishes that the nuclear dimension of these algebras is 1 under specific conditions and provides a detailed description of the space cover, extending previous results.

Findings

Nuclear dimension of $\, ext{O}_X$ is 1 for minimal shift spaces with finitely many left special elements.

Provides a detailed description of the cover of $X$ that leads to an exact sequence.

Recovers an earlier exact sequence by Carlsen and Eilers.

Abstract

For every one-sided shift space $X$ over a finite alphabet, left special elements are those points in $X$ having at least two preimages under the shift operation. In this paper, we show that the Cuntz-Pimsner $C^*$-algebra $\mathcal{O}_X$ has nuclear dimension 1 when $X$ is minimal and the number of left special elements in $X$ is finite. This is done by describing thoroughly the cover of $X$ which also recovers an exact sequence, discovered before by T. Carlsen and S. Eilers.