The noncommutative geometry of k-graph C*-algebras

This paper is comprised of two related parts. First we discuss which k-graph algebras have faithful gauge invariant traces, where the gauge action of T^k is the canonical one. We give a sufficient condition for the existence of such a trace, identify the C*-algebras of k-graphs satisfying this condition up to Morita equivalence, and compute their K-theory. For k-graphs with faithful gauge invariant trace, we construct a smooth (k,infinity)-summable semifinite spectral triple. We use the semifinite local index theorem to compute the pairing with K-theory. This numerical pairing can be obtained by applying the trace to a KK-pairing with values in the K-theory of the fixed point algebra of the T^k action. As with graph algebras, the index pairing is an invariant for a finer structure than the isomorphism class of the algebra.