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The noncommutative geometry of k-graph C*-algebras

Journal Article


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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,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.

Publication Date


  • 2008

Citation


  • Pask, D., Rennie, A. & Sims, A. (2008). The noncommutative geometry of k-graph C*-algebras. Journal of K-theory, 1 (2), 259-304.

Scopus Eid


  • 2-s2.0-85012459069

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=11308&context=infopapers

Ro Metadata Url


  • http://ro.uow.edu.au/infopapers/3965

Number Of Pages


  • 45

Start Page


  • 259

End Page


  • 304

Volume


  • 1

Issue


  • 2

Place Of Publication


  • http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=1891148&fulltextType=RA&fileId=S1755069607000138vvvvvvvvvvvvhttp://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=1891148&fulltextType=RA&fileId=S1755069607000138http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=1891148&fulltextType=RA&fileId=S1755069607000138http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=1891148&fulltextType=RA&fileId=S1755069607000138http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=1891148&fulltextType=RA&fileId=S1755069607000138http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=1891148&fulltextType=RA&fileId=S1755069607000138http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=1891148&fulltextType=RA&fileId=S1755069607000138http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=1891148&fulltextType=RA&fileId=S1755069607000138

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,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.

Publication Date


  • 2008

Citation


  • Pask, D., Rennie, A. & Sims, A. (2008). The noncommutative geometry of k-graph C*-algebras. Journal of K-theory, 1 (2), 259-304.

Scopus Eid


  • 2-s2.0-85012459069

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=11308&context=infopapers

Ro Metadata Url


  • http://ro.uow.edu.au/infopapers/3965

Number Of Pages


  • 45

Start Page


  • 259

End Page


  • 304

Volume


  • 1

Issue


  • 2

Place Of Publication


  • http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=1891148&fulltextType=RA&fileId=S1755069607000138vvvvvvvvvvvvhttp://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=1891148&fulltextType=RA&fileId=S1755069607000138http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=1891148&fulltextType=RA&fileId=S1755069607000138http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=1891148&fulltextType=RA&fileId=S1755069607000138http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=1891148&fulltextType=RA&fileId=S1755069607000138http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=1891148&fulltextType=RA&fileId=S1755069607000138http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=1891148&fulltextType=RA&fileId=S1755069607000138http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=1891148&fulltextType=RA&fileId=S1755069607000138