Partial actions and subshifts

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.