On certain properties of Cuntz--Krieger type algebras

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.