Spectral triples for higher-rank graph $C^*$-algebras

  • Carla Farsi
  • Elizabeth Gillaspy
  • Antoine Julien
  • Sooran Kang
  • Judith Packer


In this note, we present a new way to associate a spectral triple to the noncommutative $C^*$-algebra $C^*(\Lambda )$ of a strongly connected finite higher-rank graph Λ. Our spectral triple builds on an approach used by Consani and Marcolli to construct spectral triples for Cuntz-Krieger algebras. We prove that our spectral triples are intimately connected to the wavelet decomposition of the infinite path space of Λ which was introduced by Farsi, Gillaspy, Kang, and Packer in 2015. In particular, we prove that the wavelet decomposition of Farsi et al. describes the eigenspaces of the Dirac operator of our spectral triple. The paper concludes by discussing other properties of the spectral triple, namely, θ-summability and Brownian motion.


Carlsen, T. M., Kang, S., Shotwell, J., and Sims, A., The primitive ideals of the Cuntz-Krieger algebra of a row-finite higher-rank graph with no sources, J. Funct. Anal. 266 (2014), no. 4, 2570–2589. https://doi.org/10.1016/j.jfa.2013.08.029

Christensen, E., Ivan, C., and Lapidus, M. L., Dirac operators and spectral triples for some fractal sets built on curves, Adv. Math. 217 (2008), no. 1, 42–78. https://doi.org/10.1016/j.aim.2007.06.009

Clark, L. O., an Huef, A., and Sims, A., AF-embeddability of $2$-graph algebras and quasidiagonality of $k$-graph algebras, J. Funct. Anal. 271 (2016), no. 4, 958–991. https://doi.org/10.1016/j.jfa.2016.04.024

Connes, A., Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994.

Consani, C. and Marcolli, M., Noncommutative geometry, dynamics, and ∞-adic Arakelov geometry, Selecta Math. (N.S.) 10 (2004), no. 2, 167–251. https://doi.org/10.1007/s00029-004-0369-3

Davidson, K. R. and Yang, D., Periodicity in rank $2$ graph algebras, Canad. J. Math. 61 (2009), no. 6, 1239–1261. https://doi.org/10.4153/CJM-2009-058-0

Farsi, C., Gillaspy, E., Julien, A., Kang, S., and Packer, J., Spectral triples and wavelets for higher-rank graphs, J. Math. Anal. Appl. 482 (2020), no. 2, art. no. 123572, 39 pp. https://doi.org/10.1016/j.jmaa.2019.123572

Farsi, C., Gillaspy, E., Kang, S., and Packer, J., Wavelets and graph $C^*$-algebras, in “Excursions in harmonic analysis. Vol. 5”, Appl. Numer. Harmon. Anal., Birkhäuser/Springer, Cham, 2017, pp. 35–86.

Farsi, C., Gillaspy, E., Kang, S., and Packer, J. A., Separable representations, KMS states, and wavelets for higher-rank graphs, J. Math. Anal. Appl. 434 (2016), no. 1, 241–270. https://doi.org/10.1016/j.jmaa.2015.09.003

Guido, D. and Isola, T., Dimensions and singular traces for spectral triples, with applications to fractals, J. Funct. Anal. 203 (2003), no. 2, 362–400. https://doi.org/10.1016/S0022-1236(03)00230-1

Horn, R. A. and Johnson, C. R., Matrix analysis, second ed., Cambridge University Press, Cambridge, 2013.

an Huef, A., Kang, S., and Raeburn, I., Spatial realisations of KMS states on the $C^*$-algebras of higher-rank graphs, J. Math. Anal. Appl. 427 (2015), no. 2, 977–1003. https://doi.org/10.1016/j.jmaa.2015.02.045

an Huef, A., Laca, M., Raeburn, I., and Sims, A., KMS states on $C^*$-algebras associated to higher-rank graphs, J. Funct. Anal. 266 (2014), no. 1, 265–283. https://doi.org/10.1016/j.jfa.2013.09.016

an Huef, A., Laca, M., Raeburn, I., and Sims, A., KMS states on the $C^*$-algebra of a higher-rank graph and periodicity in the path space, J. Funct. Anal. 268 (2015), no. 7, 1840–1875. https://doi.org/10.1016/j.jfa.2014.12.006

Jonsson, A., Wavelets on fractals and Besov spaces, J. Fourier Anal. Appl. 4 (1998), no. 3, 329–340. https://doi.org/10.1007/BF02476031

Julien, A. and Savinien, J., Transverse Laplacians for substitution tilings, Comm. Math. Phys. 301 (2011), no. 2, 285–318. https://doi.org/10.1007/s00220-010-1150-4

Kang, S. and Pask, D., Aperiodicity and primitive ideals of row-finite $k$-graphs, Internat. J. Math. 25 (2014), no. 3, art. no. 1450022, 25 pp. https://doi.org/10.1142/S0129167X14500220

Kellendonk, J. and Savinien, J., Spectral triples from stationary Bratteli diagrams, Michigan Math. J. 65 (2016), no. 4, 715–747. https://doi.org/10.1307/mmj/1480734017

Kumjian, A. and Pask, D., Higher rank graph $C^\ast $-algebras, New York J. Math. 6 (2000), 1–20.

Kumjian, A., Pask, D., Raeburn, I., and Renault, J., Graphs, groupoids, and Cuntz-Krieger algebras, J. Funct. Anal. 144 (1997), no. 2, 505–541. https://doi.org/10.1006/jfan.1996.3001

Laca, M., Larsen, N. S., Neshveyev, S., Sims, A., and Webster, S. B. G., Von Neumann algebras of strongly connected higher-rank graphs, Math. Ann. 363 (2015), no. 1-2, 657–678. https://doi.org/10.1007/s00208-015-1187-y

Lapidus, M. L., Analysis on fractals, Laplacians on self-similar sets, noncommutative geometry and spectral dimensions, Topol. Methods Nonlinear Anal. 4 (1994), no. 1, 137–195. https://doi.org/10.12775/TMNA.1994.025

Marcolli, M. and Paolucci, A. M., Cuntz-Krieger algebras and wavelets on fractals, Complex Anal. Oper. Theory 5 (2011), no. 1, 41–81. https://doi.org/10.1007/s11785-009-0044-y

Palmer, I. C., Riemannian geometry of compact metric spaces, Ph.D. thesis, Georgia Institute of Technology, 2010.

Pask, D., Rennie, A., and Sims, A., The noncommutative geometry of $k$-graph $C^*$-algebras, J. K-Theory 1 (2008), no. 2, 259–304. https://doi.org/10.1017/is007011015jkt013

Pask, D., Rennie, A., and Sims, A., Noncommutative manifolds from graph and $k$-graph $C^\ast $-algebras, Comm. Math. Phys. 292 (2009), no. 3, 607–636. https://doi.org/10.1007/s00220-009-0926-x

Pearson, J. and Bellissard, J., Noncommutative Riemannian geometry and diffusion on ultrametric Cantor sets, J. Noncommut. Geom. 3 (2009), no. 3, 447–480. https://doi.org/10.4171/JNCG/43

Raeburn, I., Sims, A., and Yeend, T., The $C^*$-algebras of finitely aligned higher-rank graphs, J. Funct. Anal. 213 (2004), no. 1, 206–240. https://doi.org/10.1016/j.jfa.2003.10.014

Robertson, D. I. and Sims, A., Simplicity of $C^\ast $-algebras associated to higher-rank graphs, Bull. Lond. Math. Soc. 39 (2007), no. 2, 337–344. https://doi.org/10.1112/blms/bdm006

Robertson, G. and Steger, T., Affine buildings, tiling systems and higher rank Cuntz-Krieger algebras, J. Reine Angew. Math. 513 (1999), 115–144. https://doi.org/10.1515/crll.1999.057

Strichartz, R. S., Construction of orthonormal wavelets, in “Wavelets: mathematics and applications”, Stud. Adv. Math., CRC, Boca Raton, FL, 1994, pp. 23–50.
How to Cite
Farsi, C., Gillaspy, E., Julien, A., Kang, S., & Packer, J. (2020). Spectral triples for higher-rank graph $C^*$-algebras. MATHEMATICA SCANDINAVICA, 126(2), 321-338. https://doi.org/10.7146/math.scand.a-119260