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

http://dx.doi.org/10.4153/CMB-2002-035-6
Canad. Math. Bull. 45(2002), 321-336

  Published:2002-09-01
 Printed: Sep 2002

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~$E$.