The dynamics of a one-sided subshift $\mathsf{ X}$ can be modeled by a set of partially defined bijections. From this data we define an inverse semigroup $\mathcal{ S}_{ \mathsf{ X}}$ and show that it has many interesting properties. We prove that the Carlsen-Matsumoto C*-algebra $\mathcal{ O}_\mathsf{ X}$ associated to $\mathsf{ X}$ is canonically isomorphic to Exel's tight C*-algebra of $\mathcal{ S}_{ \mathsf{ X}}$. As one consequence, we obtain that $\mathcal{ O}_\mathsf{ X}$ can be written as a partial crossed product of a commutative C*-algebra by a countable group.
Categories: math.OA math.CT math.DS