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$C^{\ast}$Algebras of Infinite Graphs and CuntzKrieger Algebras


Published:20020901
Printed: Sep 2002
Abstract
The CuntzKrieger 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
CuntzKrieger algebra $\mathcal{O}_B$ where $B$ is the vertex matrix
of~$E$.