$C^*$-algebras arising from substitutions

TL;DR

This paper constructs a new $C^*$-algebra from primitive substitutions, demonstrating its simplicity, pure infiniteness, and connections to known algebras, with computed $K$-groups.

Contribution

It introduces a novel $C^*$-algebra associated with primitive substitutions, expanding the understanding of their algebraic and topological properties.

Findings

The algebra is simple and purely infinite.

It contains the Cuntz-Krieger algebra and crossed product algebra.

The $K$-groups are explicitly calculated.

Abstract

In this paper, we introduce a $C^{\ast}$-algebra associated with a proper primitive substitution. We show that the $C^{\ast}$-algebra is simple and purely infinite and contains the associated Cuntz-Krieger algebra and the crossed product $C^{\ast}$-algebra of the corresponding Cantor minimal system. We calculate the $K$-groups.