On certain properties of Cuntz--Krieger type algebras

TL;DR

This paper investigates properties of Cuntz--Krieger type algebras, providing conditions for pure infiniteness, analyzing their ideal structure, and demonstrating nuclearity through crossed product representations, with applications to various algebra classes.

Contribution

It establishes a necessary and sufficient condition for pure infiniteness and proves nuclearity by representing the algebra as a crossed product, extending understanding of Cuntz--Krieger type algebras.

Findings

Identified a condition for pure infiniteness.

Studied the ideal structure of these algebras.

Proved nuclearity via crossed product representation.

Abstract

The note presents a further study of the class of Cuntz--Krieger type algebras. A necessary and sufficient condition is identified that ensures that the algebra is purely infinite, the ideal structure is studied, % and applied to semigraph algebras, and nuclearity is proved by presenting the algebra as a crossed product of an AF-algebra by an abelian group. The results are applied to examples of Cuntz--Krieger type algebras, such as higher rank semigraph $C^*$-algebras and higher rank Exel-Laca algebras.