## Annals of Functional Analysis

- Ann. Funct. Anal.
- Volume 8, Number 3 (2017), 386-397.

### On certain properties of Cuntz–Krieger-type algebras

Bernhard Burgstaller and D. Gwion Evans

#### Abstract

This 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 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 ${C}^{\ast}$-algebras and higher-rank Exel–Laca algebras.

#### Article information

**Source**

Ann. Funct. Anal., Volume 8, Number 3 (2017), 386-397.

**Dates**

Received: 29 April 2016

Accepted: 6 November 2016

First available in Project Euclid: 9 May 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.afa/1494295270

**Digital Object Identifier**

doi:10.1215/20088752-2017-0004

**Mathematical Reviews number (MathSciNet)**

MR3690001

**Zentralblatt MATH identifier**

1380.46038

**Subjects**

Primary: 46L05: General theory of $C^*$-algebras

Secondary: 46L55: Noncommutative dynamical systems [See also 28Dxx, 37Kxx, 37Lxx, 54H20]

**Keywords**

Cuntz–Krieger semigraph algebra ideal purely infinite crossed product

#### Citation

Burgstaller, Bernhard; Evans, D. Gwion. On certain properties of Cuntz–Krieger-type algebras. Ann. Funct. Anal. 8 (2017), no. 3, 386--397. doi:10.1215/20088752-2017-0004. https://projecteuclid.org/euclid.afa/1494295270