A class of simple $C^*$-algebras arising from certain nonsofic subshifts

TL;DR

This paper introduces a new class of subshifts $Z_N$ with associated $C^*$-algebras that are simple, purely infinite, and not stably isomorphic to known Cuntz-Krieger or Cuntz algebras, with detailed entropy and KMS-state analysis.

Contribution

It constructs the first examples of subshifts whose associated $C^*$-algebras are not stably isomorphic to any Cuntz-Krieger or Cuntz algebra, expanding the understanding of such algebras.

Findings

The $C^*$-algebras ${ mf O}_{Z_N}$ are simple and purely infinite.

Topological entropy of $Z_N$ is explicitly computed.

KMS-states exist uniquely at inverse temperatures equal to the topological entropy.

Abstract

We present a class of subshifts $Z_N, N = 1,2,...$ whose associated $C^*$-algebras ${\cal O}_{Z_N}$ are simple, purely infinite and not stably isomorphic to any Cuntz-Krieger algebra nor to Cuntz algebra. The class of the subshifts is the first examples whose associated $C^*$-algebras are not stably isomorphic to any Cuntz-Krieger algebra nor to Cuntz algebra. The subshifts $Z_N$ are coded systems whose languages are context free. We compute the topological entropy for the subshifts and show that KMS-state for gauge action on the associated $C^*$-algebra ${\cal O}_{Z_N}$ exists if and only if the logarithm of the inverse temperature is the topological entropy for the subshift $Z_N$, and the corresponding KMS-state is unique.