Inverse semigroups associated to subshifts

Abstract

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.

arXiv comment24 pages. The results in the published version are not correct as stated, but become correct when one assumes X satisfies Matsumoto's condition (I). A corrigendum has been submitted to J. Algebra, and has been reproduced one pages 20-22 of this entry. Some references fixed between last version and this