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# $C^{\ast}$-Algebras of Infinite Graphs and Cuntz-Krieger Algebras

Published:2002-09-01
Printed: Sep 2002
• Berndt Brenken
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## Abstract

The Cuntz-Krieger algebra $\mathcal{O}_B$ is defined for an arbitrary, possibly infinite and infinite valued, matrix $B$. A graph $C^{\ast}$-algebra $G^{\ast} (E)$ is introduced for an arbitrary directed graph $E$, and is shown to coincide with a previously defined graph algebra $C^{\ast} (E)$ if each source of $E$ emits only finitely many edges. Each graph algebra $G^{\ast} (E)$ is isomorphic to the Cuntz-Krieger algebra $\mathcal{O}_B$ where $B$ is the vertex matrix of~$E$.
 MSC Classifications: 46LXX - unknown classification 46LXX05C50 - Graphs and linear algebra (matrices, eigenvalues, etc.)