Abstract
Fuzzy tori are finite dimensional C*-algebras endowed with an appropriate notion of noncommutative geometry inherited from an ergodic action of a finite closed subgroup of the torus, which are meant as finite dimensional approximations of tori and more generally, quantum tori. A mean to specify the geometry of a noncommutative space is by constructing over it a spectral triple. We prove in this paper that we can construct spectral triples on fuzzy tori which, as the dimension grows to infinity and under other natural conditions, converge to a natural spectral triple on quantum tori, in the sense of the spectral propinquity. This provides a formal assertion that indeed, fuzzy tori approximate quantum tori, not only as quantum metric spaces, but as noncommutative differentiable manifolds—including convergence of the state spaces as metric spaces and of the quantum dynamics generated by the Dirac operators of the spectral triples, in an appropriate sense.
Similar content being viewed by others
References
Aguilar, K., Kaad, J.: The Podleś sphere as a spectral metric space. J. Geom. Phys. 133, 260–278 (2018)
Aguilar, K., Latrémolière, F.: Quantum ultrametrics on AF algebras and the Gromov–Hausdorff propinquity. Studia Math. 231(2), 149–194 (2015). arXiv:1511.07114
Barrett, J.: Matrix geometries and fuzzy spaces as finite spectral triples. J. Math. Phys. 56(8), 082301 (2015)
Bhowmick, J., Voigt, C., Zacharias, J.: Compact quantum metric spaces from quantum groups of rapid decay. Submitted (2014). arXiv:1406.0771
Bratteli, O.: Inductive limits of finite dimensional \(C^\ast \)-algebras. Trans. Am. Math. Soc. 171, 195–234 (1972)
Christ, M., Rieffel, M.A.: Nilpotent group \(C^\ast \)-algebras-algebras as compact quantum metric spaces. Can. Math. Bull. 60(1), 77–94 (2017). arXiv:1508.00980
Christensen, E., Ivan, C., Lapidus, M.: Dirac operators and spectral triples for some fractal sets built on curves. Adv. Math. 217(1), 42–78 (2008)
Connes, A.: C*-algébres et géométrie differentielle. C. R. de l’Academie des Sciences de Paris (Series A-B), 290 (1980)
Connes, A.: Compact metric spaces, Fredholm modules and hyperfiniteness. Ergodic Theory Dyn. Syst. 9(2), 207–220 (1989)
Connes, A.: Noncommutative Geometry. Academic Press, San Diego (1994)
Connes, A., Douglas, M., Schwarz, A.: Noncommutative geometry and matrix theory: compactification on tori. JHEP (1998). arXiv:Hep-th/9711162
Dąbrowski, L., Sitarz, A.: Curved noncommutative torus and Gauß–Bonnet. J. Math. Phys. 54, 013518 (2013)
Dąbrowski, L., Sitarz, A.: An asymmetric noncommutative torus. SIGMA 11, 11 (2015). arXiv:1406.4645
Davidson, K.R.: C*-Algebras by Example. Fields Institute Monographs. American Mathematical Society, Providence (1996)
Dobrushin, R.L.: Prescribing a system of random variables by conditional probabilities. Theory Probab. Appl. 15(3), 459–486 (1970)
Edwards, D.A.: The structure of superspace. In: Studies in Topology (Proceedings Conference, University of North Carolina, Charlotte, NC, 1974; dedicated to Math. Sect. Polish Acad. Sci.), pp. 121–133 (1975)
Gabriel, O., Grensing, M.: Ergodic actions and spectral triples. J. Oper. Theory 76(2), 307–334 (2016)
Glimm, J.: On a certain class of operator algebras. Trans. Am. Math. Soc. 95, 318–340 (1960)
Gromov, M.: Groups of polynomial growth and expanding maps. Publ. Math. Inst. Hautes Études Sci. 53, 53–78 (1981)
Gromov, M.: Metric Structures for Riemannian and Non-Riemannian Spaces. Progress in Mathematics. Birkhäuser, Basel (1999)
Hausdorff, F.: Grundzüge der Mengenlehre. Verlag Von Veit und Comp. (1914)
Hawkins, A., Skalski, A., White, S., Zacharias, J.: On spectral triples on crossed products arising from equicontinuous actions. Math. Scand. 113, 262–291 (2013). arXiv:1103.6199
Kantorovich, L.V.: On one effective method of solving certain classes of extremal problems. Dokl. Akad. Nauk. USSR 28, 212–215 (1940)
Kantorovich, L.V., Rubinstein, G.S.: On the space of completely additive functions. Vestnik Leningrad Univ. Ser. Mat. Mekh. i Astron. 13(7), 52–59 (1958). (in Russian)
Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1995)
Kimura, Y.: Noncommutative gauge theories on fuzzy sphere and fuzzy torus from matrix model. Nuclear Phys. B 604(1–2), 121–147 (2001)
Kleppner, A.: Multipliers on Abelian groups. Math. Ann. 158, 11–34 (1965)
Kleppner, A.: Continuity and measurability of multiplier and projective representations. J. Funct. Anal. 17, 214–226 (1974)
Landry, T., Lapidus, M., Latrémoliére, F.: Metric approximations of the spectral triple on the sierpinky gasket and other fractals. Adv. Math 385, 43 (2021). https://doi.org/10.1016/j.aim.2021.107771
Latrémoliére, F.: Approximation of the quantum tori by finite quantum tori for the quantum Gromov–Hausdorff distance. J. Funct. Anal. 223, 365–395 (2005). arXiv:math/0310214
Latrémoliére, F.: Bounded-Lipschitz distances on the state space of a C*-algebra. Taiwan. J. Math. 11(2), 447–469 (2007). arXiv:math/0510340
Latrémoliére, F.: Quantum locally compact metric spaces. J. Funct. Anal. 264(1), 362–402 (2013). arXiv:1208.2398
Latrémoliére, F.: Convergence of fuzzy tori and quantum tori for the quantum Gromov–Hausdorff propinquity: an explicit approach. Münster J. Math. 8(1), 57–98 (2015). arXiv:1312.0069
Latrémoliére, F.: Curved noncommutative tori as Leibniz compact quantum metric spaces. J. Math. Phys. 56(12), 123503 (2015). arXiv:1507.08771
Latrémoliére, F.: The dual Gromov–Hausdorff propinquity. J. Math. Pures Appl. 103(2), 303–351 (2015). arXiv:1311.0104
Latrémoliére, F.: Equivalence of quantum metrics with a common domain. J. Math. Anal. Appl. 443, 1179–1195 (2016). arXiv:1604.00755
Latrémolière, F.: The quantum Gromov–Hausdorff propinquity. Trans. Am. Math. Soc. 368(1), 365–411 (2016)
Latrémoliére, F.: A compactness theorem for the dual Gromov–Hausdorff propinquity. Indiana Univ. Math. J. 66(5), 1707–1753 (2017). arXiv:1501.06121
Latrémoliére, F.: The triangle inequality and the dual Gromov–Hausdorff propinquity. Indiana Univ. Math. J. 66(1), 297–313 (2017). arXiv:1404.6633
Latrémoliére, F.: The dual-modular Gromov–Hausdorff propinquity and completeness. J. Noncomm. Geom 15(1), 347–398 (2021). https://doi.org/10.4171/jncg/414
Latrémoliére, F.: The Gromov–Hausdorff propinquity for metric spectral triples. Submitted arXiv:1811.10843 (2018)
Latrémoliére, F.: Actions of categories by Lipschitz morphisms on limits for the Gromov–Hausdorff propinquity. J. Geom. Phys. 146, 103481 (2019). arXiv:1708.01973
Latrémoliére, F.: Convergence of Cauchy sequences for the covariant Gromov–Hausdorff propinquity. J. Math. Anal. Appl. 469(1), 378–404 (2019). arXiv:1806.04721
Latrémoliére, F.: The modular Gromov–Hausdorff propinquity. Diss. Math. 544, 1–70 (2019). arXiv:1608.04881
Latrémoliére, F.: Convergence of Heisenberg modules for the modular Gromov–Hausdorff propinquity. J. Oper. Theory 84(1), 211–237 (2020)
Latrémoliére, F.: The covariant Gromov–Hausdorff propinquity. Studia Math. 251(2), 135–169 (2020). arXiv:1805.11229
Latrémoliére, F.: Heisenberg modules over quantum \(2\)-tori are metrized quantum vector bundles. Can. J. Math 72(4), 1044–1081 (2020). arXiv:1703.07073
Latrémoliére, F., Packer, J.: Noncommutative solenoids and the Gromov–Hausdorff propinquity. Proc. Am. Math. Soc. 145(5), 1179–1195 (2017). arXiv:1601.02707
Li, H.: Compact quantum metric space and ergodic actions of compact quantum groups. J. Funct. Anal. 256(10), 3368–3408 (2009). https://doi.org/10.1016/j.jfa.2008.09.009
Li, H.: Metric aspects of noncommutative homogenous space. J. Funct. Anal. 257(7), 2325–2350 (2009). arXiv:0810.4694
Mackey, G.W.: Unitary representations of group extensions, i. Acta Math. 99, 265–311 (1958)
McShane, E.J.: Extension of range of functions. Bull. Am. Math. Soc. 40(12), 825–920 (1934)
Ozawa, N., Rieffel, M.A.: Hyperbolic group \(C^{\ast }\)-algebras and free product \(C^{\ast }\)-algebras as compact quantum metric spaces. Can. J. Math. 57, 1056–1079 (2005). arXiv:math/0302310
Reed, M., Simon, B.: Functional Analysis. Methods of Modern Mathematical Physics. Academic Press, San Diego (1980)
Renault, J.: A Groupoid Approach to C*-Algebras. Lecture Notes in Mathematics, vol. 793. Springer, Berlin (1980)
Rieffel, M.A.: C*-algebras associated with irrational rotations. Pac. J. Math. 93, 415–429 (1981)
Rieffel, M.A.: Non-commutative tori—a case study of non-commutative differentiable manifolds. Contemp. Math. 105, 191–211 (1990)
Rieffel, M.A.: Metrics on states from actions of compact groups. Doc. Math. 3, 215–229 (1998). arXiv:math/9807084
Rieffel, M.A.: Metrics on state spaces. Doc. Math. 4, 559–600 (1999). arXiv:math/9906151
Rieffel, M.A.: Group \(C^{\ast }\)-algebras as compact quantum metric spaces. Doc. Math. 7, 605–651 (2002). arXiv:math/0205195
Rieffel, M.A.: Gromov–Hausdorff distance for quantum metric spaces. Mem. Am. Math. Soc. 168(796), 1–65 (2004). arXiv:math/0011063
Rieffel, M.A.: Matrix algebras converge to the sphere for quantum Gromov–Hausdorff distance. Mem. Am. Math. Soc. 168(796), 67–91 (2004). arXiv:math/0108005
Rieffel, M.A.: Distances between matrix alegbras that converge to coadjoint orbits. Proc. Sympos. Pure Math. 81, 173–180 (2010). arXiv:0910.1968
Rieffel, M.A.: Leibniz seminorms for “Matrix algebras converge to the sphere”. Clay Math. Proc. 11, 543–578 (2010)
Rieffel, M.A.: Standard deviation is a strongly Leibniz seminorm. N. Y. J. Math. 20, 35–56 (2014). arXiv:1208.4072
Rieffel, M.A.: Matricial bridges for “matrix algebras converge to the sphere”. In: Operator Algebras and Their Applications. Contemporary Mathematics, vol. 671, pp. 209–233. American Mathematical Society, Providence, RI (2016). arXiv:1502.00329
Santhanam, T., Sinha, K.B.: Quantum mechanics in finite dimensions. Aust. J. Phys. 31, 233–238 (1978)
Schreivogl, P., Steinacker, H.: Generalized fuzzy torus and its modular properties. SIGMA 9(060), 23 (2013)
Seiberg, N., Witten, E.: String theory and noncommutative geometry. JHEP 9909(32) (1999). arXiv:hep-th/9908142
T’Hooft, G.: Determinism beneath quantum mechanics. Presentation at “Quo Vadis Quantum Mechanics?” Temple University, Philadelphia (2002). arXiv:quant-ph/0212095
Vourdas, A.: Quantum systems with finite Hilbert space. Rep. Prog. Phys. 67(4), 267–320 (2004)
Wasserstein, L.N.: Markov processes on a countable product space, describing large systems of automata. Problemy Peredachi Infomatsii 5(3), 64–73 (1969). in Russian
Weyl, H.: The Theory of Groups and Quantum Mechanics. Dover Publication, New York (1950) (Translated from the second (revised) German edition by H. P. Robertson)
Zeller-Meier, G.: Produits croisés d’une C*-algébre par un groupe d’ Automorphismes. J. Math. Pures Appl. 47(2), 101–239 (1968)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by H.-T. Yau.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
Latrémolière, F. Convergence of Spectral Triples on Fuzzy Tori to Spectral Triples on Quantum Tori. Commun. Math. Phys. 388, 1049–1128 (2021). https://doi.org/10.1007/s00220-021-04173-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-021-04173-0