Irreducible representations of one-sided subshift algebras

TL;DR

This paper constructs and analyzes representations of subshift algebras, identifying conditions for faithfulness, irreducibility, and minimal ideals, thereby advancing understanding of their algebraic structure.

Contribution

It introduces criteria for faithfulness and characterizes irreducible components of subshift algebra representations, including minimal left ideals associated with line paths.

Findings

Faithfulness of representation linked to cycles with no exits.

Irreducible components characterized and classified.

Minimal left ideals identified for line paths.

Abstract

Given a subshift over an arbitrary alphabet, we construct a representation of the associated unital algebra. We describe a criteria for the faithfulness of this representation in terms of the existence of cycles with no exits. Subsequently, we focus on describing the irreducible components of this representation and characterize equivalence between such components. Building upon these findings, we identify the minimal left ideals of a subshift algebra associated with (what we call) line paths.