Actions of symbolic dynamical systems on $C^*$-algebras II. Simplicity of $C^*$-symbolic crossed products and some examples

TL;DR

This paper investigates the simplicity of $C^*$-algebras arising from $C^*$-symbolic dynamical systems, providing criteria and examples including irrational rotation Cuntz-Krieger algebras.

Contribution

It establishes conditions for simplicity of $C^*$-algebras from symbolic dynamical systems and explores specific examples like irrational rotation Cuntz-Krieger algebras.

Findings

Simplicity conditions for $C^*$-symbolic crossed products are identified.

Examples demonstrate the application of these conditions to specific algebras.

The study extends understanding of the structure of $C^*$-algebras from symbolic dynamics.

Abstract

We have introduced a notion of $C^*$-symbolic dynamical system in [K. Matsumoto: Actions of symbolic dynamical systems on $C^*$-algebras, to appear in J. Reine Angew. Math.], that is a finite family of endomorphisms of a $C^*$-algebra with some conditions. The endomorphisms are indexed by symbols and yield both a subshift and a $C^*$-algebra of a Hilbert $C^*$-bimodule. The associated $C^*$-algebra with the $C^*$-symbolic dynamical system is regarded as a crossed product by the subshift. We will study a simplicity condition of the $C^*$-algebras of the $C^*$-symbolic dynamical systems. Some examples such as irrational rotation Cuntz-Krieger algebras will be studied.