Inverse semigroups associated to subshifts

TL;DR

This paper constructs an inverse semigroup from a subshift's dynamics, linking it to the associated C*-algebra and revealing its structure as a partial crossed product.

Contribution

It introduces a new inverse semigroup model for subshift dynamics and establishes an isomorphism with the Carlsen-Matsumoto C*-algebra, connecting algebraic and dynamical systems.

Findings

Inverse semigroup $ ext{S}_ ext{X}$ models subshift dynamics

C*-algebra $ ext{O}_ ext{X}$ is isomorphic to Exel's tight C*-algebra

$ ext{O}_ ext{X}$ can be expressed as a partial crossed product

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.