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The Noncommutative Geometry of k-graph C*-Algebras

David Pask·Adam Rennie·Aidan Sims
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Abstract

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.

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David Pask, Adam Rennie, Aidan Sims. 2007-01-17. The Noncommutative Geometry of k-graph C*-Algebras. https://arxiv.org/abs/math/0512438

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