Computing $K$-theory and $\mathrm{Ext}$ for graph $C^*$-algebras

Abstract

$K$-theory and $\mathrm{Ext}$ are computed for the $C^*$-algebra $C^*(E)$ of any countable directed graph $E$. The results generalize the $K$-theory computations of Raeburn and Szymański and the $\mathrm{Ext}$ computations of Tomforde for row-finite graphs. As a consequence, it is shown that if $A$ is a countable $\{0,1\}$ matrix and $E_A$ is the graph obtained by viewing $A$ as a vertex matrix, then $C^*(E_A)$ is not necessarily Morita equivalent to the Exel-Laca algebra $\mathcal{O}_A$.