The noncommutative geometry of graph C∗-algebras I: The index theorem

We investigate conditions on a graph C∗-algebra for the existence of a faithful semifinite trace. Using such a trace and the natural gauge action of the circle on the graph algebra, we construct a smooth (1,∞)-summable semi-finite spectral triple. The local index theorem allows us to compute the pairing with K-theory. This produces invariants in the K-theory of the fixed point algebra, and these are invariants for a finer structure than the isomorphism class of C∗(E). © 2005 Published by Elsevier Inc