Algebras of one-sided subshifts over arbitrary alphabets

TL;DR

This paper introduces two algebras linked to subshifts over any alphabet, focusing on the unital case, and characterizes conjugacy via algebraic and topological isomorphisms.

Contribution

It defines new algebraic structures for subshifts over arbitrary alphabets and characterizes conjugacy through homeomorphisms and algebra isomorphisms.

Findings

Conjugacy of subshifts corresponds to algebra isomorphisms.

Unital algebra can be realized as a groupoid algebra.

Provides a description of subshift algebras via partial skew group rings.

Abstract

We introduce two algebras associated with a subshift over an arbitrary alphabet. One is unital and the other not necessarily. We focus on the unital case and describe a conjugacy between Ott-Tomforde-Willis subshifts in terms of a homeomorphism between the Stone duals of suitable Boolean algebras, and in terms of a diagonal-preserving isomorphism of the associated unital algebras. For this, we realise the unital algebra associated with a subshift as a groupoid algebra and as a partial skew group ring.