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

Dirceu Bagio, Cristóbal Gil Canto, Daniel Gonçalves, Danilo Royer

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).

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

Abstract

Let be a commutative unital ring, a subshift, and the corresponding unital subshift algebra. We establish the reduction theorem for . As a consequence, we obtain a Cuntz-Krieger uniqueness theorem for and we show that is semiprimitive (resp. semiprime) whenever is a field (resp. a domain).
Paper Structure (7 sections, 14 theorems, 93 equations)

This paper contains 7 sections, 14 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Symbolic Dynamics
  3. The unital subshift algebra
  4. Definition and the partial skew group ring realization
  5. Relative ranges and multiplicative properties.
  6. The Reduction Theorem
  7. Some consequences of the Reduction Theorem

Key Result

Lemma 2.2

Let $\mathscr{A}$ be an alphabet and $\alpha$ and $\beta$ be two words in $\mathscr{A}^+$.

Theorems & Definitions (33)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 3.1
  • Proposition 3.2
  • Theorem 3.4
  • Lemma 3.5
  • Lemma 3.6
  • proof
  • Lemma 3.7
  • ...and 23 more