We classify the gauge-invariant ideals in the $C^*$-algebras of infinite directed graphs, and describe the quotients as graph algebras. We then use these results to identify the gauge-invariant primitive ideals in terms of the structural properties of the graph, and describe the $K$-theory of the $C^*$-algebras of arbitrary infinite graphs.
"The ideal structure of the $C\sp *$-algebras of infinite graphs." Illinois J. Math. 46 (4) 1159 - 1176, Winter 2002. https://doi.org/10.1215/ijm/1258138472