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The noncommutative geometry of k-graph C*-algebras

Published online by Cambridge University Press:  21 December 2007

David Pask ,
Adam Rennie  and
Aidan Sims
David Pask
Affiliation:
dpask@uow.edu.au School of Mathematics and Applied Statistics, Austen Keene Building (15), University of Wollongong, NSW 2522, Australia
Adam Rennie
Affiliation:
rennie@math.ku.dk Institute for Mathematical Sciences, Universitetsparken 5, DK-2100, Copenhagen, Denmark adam.rennie@edu.au Department of Mathematics, Australian National University, John Dedman Building (Building 27), ANU, Acton, Canberra 0200, Australia
Aidan Sims
Affiliation:
asims@uow.edu.au School of Mathematics and Applied Statistics, Austen Keene Building (15), University of Wollongong, NSW 2522, Australia
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Abstract

This paper is comprised of two related parts. First we discuss which k-graph algebras have faithful traces. We characterise the existence of a faithful semifinite lower-semicontinuous gauge-invariant trace on C* (Λ) in terms of the existence of a faithful graph trace on Λ.

Second, for k-graphs with faithful gauge invariant trace, we construct a smooth (k,∞)-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 Tk action. As with graph algebras, the index pairing is an invariant for a finer structure than the isomorphism class of the algebra.

Keywords

Graph algebraspectral tripleindex theoremKK-theory
Type
Research Article
Information
Journal of K-Theory , Volume 1 , Issue 2 , April 2008 , pp. 259 - 304
Copyright
Copyright © ISOPP 2008

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