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

Abstract

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

20 pages. Revised version. To appear in Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas