The Noncommutative Geometry of k-graph C*-Algebras
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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 $(k,\infty)$-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-theory
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-theory