Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-24T00:22:13.141Z Has data issue: false hasContentIssue false
  • English
  • Français

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:
[email protected] School of Mathematics and Applied Statistics, Austen Keene Building (15), University of Wollongong, NSW 2522, Australia
Adam Rennie
Affiliation:
[email protected] Institute for Mathematical Sciences, Universitetsparken 5, DK-2100, Copenhagen, Denmark [email protected] Department of Mathematics, Australian National University, John Dedman Building (Building 27), ANU, Acton, Canberra 0200, Australia
Aidan Sims
Affiliation:
[email protected] School of Mathematics and Applied Statistics, Austen Keene Building (15), University of Wollongong, NSW 2522, Australia
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

BPRS. Bates, T., Pask, D., Raeburn, I. and Szymanski, W., The C*-Algebras of Row -Finite Graphs, New York J. Math. 6 (2000), 307–324. BCPRSW (line 19 page 39)Google Scholar
BCPRSW. Benemeur, M-T., Carey, A., Phillips, J., Rennie, A., Sukochev, F., Woj-ciechowski, K., An Analytic Approach to Spectral Flow in von Neumann Algebras, in ‘Analysis, Geometry and Topology of Elliptic Operators-Papers in Honour of K. P. Wojciechowski’, World Scientific, 2006, 297–352CrossRefGoogle Scholar
CPS2. Carey, A., Phillips, J. and Sukochev, F., Spectral Flow and Dixmier Traces Advances in Mathematics, 173 (2003), 68–113CrossRefGoogle Scholar
CPRS1. Carey, A., Phillips, J., Rennie, A. and Sukochev, F., The Hochschild Class of the Chern Character of Semifinite Spectral Triples, Journal of Functional Analysis 213 (2004), 111–153CrossRefGoogle Scholar
CPRS2. Carey, A., Phillips, J., Rennie, A. and Sukochev, F., The Local Index Theorem in Semifinite von Neumann Algebras I: Spectral Flow, Advances in Mathematics 202 (2006), 451–516CrossRefGoogle Scholar
CPRS3. Carey, A., Phillips, J., Rennie, A. and Sukochev, F., The Local Index Theorem in Semifinite von Neumann Algebras II: The Even Case, Advances in Mathematics 202 (2006) 517–554CrossRefGoogle Scholar
CPRS4. Carey, A., Phillips, J., Rennie, A., Sukochev, F. The Chern Character ofSemifinite Spectral Triples, math.OA/0611227Google Scholar
C. Connes, A., Noncommutative Geometry, Academic Press, 1994Google Scholar
CM. Connes, A. and Moscovici, H., The Local Index Formula in Noncommutative Geometry, GAFA 5 (1995), 174–243Google Scholar
D. Dixmier, J., Von Neumann Algebras, North-Holland, 1981Google Scholar
E. Evans, D. G., On Higher-rank Graph C*-Algebras, Ph.D. Thesis, Univ. Wales, 2002Google Scholar
FK. Fack, T. and Kosaki, H., Generalised s-numbers of τ-measurable operators, Pacific J. Math. 123 (1986), 269–300CrossRefGoogle Scholar
FS. Fowler, N. J. and Sims, A., Product systems over right-angled Artin semigroups, Trans. Amer. Math. Soc. 354(2002), 1487–1509CrossRefGoogle Scholar
GGISV. Gayral, V., Gracia-Bondía, J.M., Iochum, B., Schücker, T. and Varilly, J.C., Moyal Planes are Spectral Triples, Comm. Math. Phys. 246 (2004), 569–623CrossRefGoogle Scholar
GVF. Gracia-Bondía, J. M., Varilly, J. C. and Figueroa, H., Elements of Noncommutative Geometry, Birkhauser, Boston, 2001CrossRefGoogle Scholar
HR. Higson, N. and Roe, J., Analytic K-Homology, Oxford University Press, 2000Google Scholar
H. Hjelmborg, J. v.B., Purely infinite and stable C*-algebras of graphs and dynamical systems, Ergod. Th. & Dynam. Sys. 21 (2001), 1789–1808CrossRefGoogle Scholar
KNR. Kaad, J., Nest, R., Rennie, A., KK-theory and spectral flow in von Neumann algebras, math.OA/0701326Google Scholar
K. Kasparov, G. G., The Operator K-Functor and Extensions of C*-Algebras, Math. USSR. Izv. 16 (1981), 513–572CrossRefGoogle Scholar
KP. Kumjian, A. and Pask, D., Higher rank graph C*-algebras, New York J. Math. 6 (2000), 1–20Google Scholar
KPR. Kumjian, A., Pask, D. and Raeburn, I., Cuntz-Krieger Algebras of Directed Graphs, Pacific J. Math. 184 (1998), 161–174CrossRefGoogle Scholar
KPRR. Kumjian, A., Pask, D., Raeburn, I. and Renault, J., Graphs, Groupoids and Cuntz-Krieger Algebras, J. Funct. Anal. 144(1997), 505–541CrossRefGoogle Scholar
L. Lance, E. C., Hilbert C*-Modules, Cambridge University Press, Cambridge, 1995CrossRefGoogle Scholar
La. Landi, G., An Introduction to Noncommutative Spaces and their Geometries, Lecture Notes in Physics, New Series 51, Springer-Verlag, Berlin, 1997Google Scholar
M. Mallios, A., Topological Algebras, Selected Topics, Elsevier Science Publishers B.V., 1986Google Scholar
PR. Pask, D. and Raeburn, I., On the K-Theory of Cuntz-Krieger Algebras, Publ. RIMS, Kyoto Univ., 32(1996), 415–443Google Scholar
PRRS. Pask, D., Raeburn, I., Rørdam, M. and Sims, A., Rank-2 graphs whose C*-algebras are direct limits of circle algebras, J. Funct. Anal. 239 (2006), 137–178CrossRefGoogle Scholar
PRen. Pask, D. and Rennie, A., The Noncommutative Geometry of Graph C*-Algebras I: The Index Theorem, J. Funct. Anal. 233 (2006), 92–134CrossRefGoogle Scholar
PhR. Phillips, J. and Raeburn, I., An Index Theorem for Toeplitz Operators with Noncommutative Symbol Space, J. Funct. Anal. 120 (1994), 239–263CrossRefGoogle Scholar
RSY. Raeburn, I., Sims, A., and Yeend, T., Higher-rank graphs and their C* -algebras Proc. Edinb. Math. Soc. 46 (2003), 99–115CrossRefGoogle Scholar
RW. Raeburn, I. and Williams, D. P., Morita Equivalence and Continuous-Trace C*-Algebras, Math. Surveys & Monographs, vol. 60, Amer. Math. Soc., Providence, 1998CrossRefGoogle Scholar
RSz. Raeburn, I. and Szymanski, W., Cuntz-Krieger Algebras of Infinite Graphs and Matrices, Trans. Amer. Math. Soc. 356 (2004), 39–59CrossRefGoogle Scholar
R1. Rennie, A., Smoothness and Locality for Nonunital Spectral Triples, K-theory, 28 (2003), 127–165CrossRefGoogle Scholar
R2. Rennie, A., Summability for Nonunital Spectral Triples, K-theory 31 (2004), 71–100CrossRefGoogle Scholar
S. Schweitzer, Larry B., A Short Proof that Mn(A) is local if A is Local and Fréchet, Int. J. math. 3 (1992), 581–589CrossRefGoogle Scholar
Si. Sims, A., Gauge-invariant ideals in the C*-algebras of finitely aligned higher-rank graphs, Canad. J. math, Canad. J. Math. 58 (2006), 1268–1290CrossRefGoogle Scholar
T. Tomforde, M., The ordered K0-group of a graph C*-algebra, C. R. Math. Acad. Sci. Soc. R. Can. 25 (2003), 19–25Google Scholar