2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/122042This 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 $(k,\infty)$-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.38 pages, some pictures drawn in picTeX Some minor technical revisions. Material has been reorganised with detailed discussion of k-graphs admitting graph traces shortened and moved to an appendix. This version to appear in K-theoryOperator AlgebrasK-Theory and Homology46L05text