Partial actions and subshifts

Abstract

Given a finite alphabet Λ, and a not necessarily finite type subshift X⊆Λ∞, we introduce a partial action of the free group F(Λ) on a certain compactification ΩX of X, which we call the spectral partial action.The space ΩX has already appeared in many papers in the subject, arising as the spectrum of a commutative C*-algebra usually denoted by DX. A good understanding of DX is crucial for the study of C*-algebras related to subshifts, and since the descriptions given of ΩX 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. Thus the spectral partial action permits us to endow the Carlsen–Matsumoto C*-algebra OX with a partial crossed product structure. We combine this with our characterization of the dynamical properties of the spectral partial action, in order to treat the problem of simplicity of OX, considered earlier by several authors. As a new advance, we are able to give necessary and sufficient conditions for OX to be simple, without imposing any restriction on X, and this is done in terms of transparent “graphical” properties of X. As a by-product of our partial action approach, we easily recover some known facts on OX, putting them more in line with mainstream techniques used to treat similar C*-algebras.