The reduction theorem for algebras of one-sided subshifts over arbitrary alphabets

TL;DR

This paper proves a reduction theorem for subshift algebras over arbitrary alphabets, leading to a uniqueness theorem and properties like semiprimitivity depending on the base ring.

Contribution

It establishes the reduction theorem for unital subshift algebras over arbitrary alphabets, extending previous results and deriving new algebraic properties.

Findings

Reduction theorem for subshift algebras proved.

Cuntz-Krieger uniqueness theorem established.

Algebra is semiprimitive over fields and semiprime over domains.

Abstract

Let $R$ be a commutative unital ring, $\textsf{X}$ a subshift, and $\widetilde{\mathcal{A}}_R(\textsf{X})$ the corresponding unital subshift algebra. We establish the reduction theorem for $\widetilde{\mathcal{A}}_R(\textsf{X})$. As a consequence, we obtain a Cuntz-Krieger uniqueness theorem for $\widetilde{\mathcal{A}}_R(\textsf{X})$ and we show that $\widetilde{\mathcal{A}}_R(\textsf{X})$ is semiprimitive (resp. semiprime) whenever $R$ is a field (resp. a domain).