Partial actions and subshifts

TL;DR

This paper introduces a detailed model for the spectral partial action of the free group on a compactification of a subshift, facilitating a deeper understanding of associated C*-algebras and their properties.

Contribution

It provides a clear, natural model for the spectral partial action on the space $\u25b6_X$, enabling detailed structural analysis and application to C*-algebras linked to subshifts.

Findings

A new model for the spectral partial action $_X$ is developed.

Conditions for minimality of the Carlsen-Matsumoto C*-algebra are characterized.

The model allows for more natural proofs of known results and generalizations.

Abstract

Given a finite alphabet $\Lambda$, and a not necessarily finite type subshift $X\subseteq \Lambda^\infty$, we introduce a partial action of the free group $F(\Lambda)$ on a certain compactification $\Omega_X$ of $X$, which we call the spectral partial action. The space $\Omega_X$ has already appeared in many papers in the subject, arising as the spectrum of a commutative C*-algebra usually denoted by ${\cal D}_X$. Since the descriptions given of $\Omega_X$ in the literature are often somewhat terse and obscure, one of our main goals is to present a sensible model for it which allows for a detailed study of its structure, as well as of the spectral partial action, from various points of view, including topological freeness and minimality. We then apply our results to study certain C*-algebras associated to $X$, introduced by Matsumoto and Carlsen. Most of the results we prove are already well known, but our proofs are hoped to be more natural and more in line with mainstream techniques used to treat similar C*-algebras. The clearer understanding of $\Omega_X$ provided by our model in turn allows for a fine tuning of some of these results, including a necessary and sufficient condition for the minimality of the Carlsen-Matsumoto C*-algebra ${\cal O}_X$, generalizing a similar result of Thomsen.