Spectral Triples and Quantum Isometry Groups on Group \(C^{*}\)-Algebras

  • Debashish Goswami
  • Jyotishman Bhowmick
Part of the Infosys Science Foundation Series book series (ISFS)


We discuss the quantum isometry group of the reduced \( C^* \) algebra of a finitely generated discrete group. The relevant spectral triple are the ones defined by Connes which arise from length functions. We prove the existence of quantum isometry groups for such spectral triples using results of Sect.  3.4 of Chap. 3 and then present detailed computation for a number of interesting examples.


Dirac Operator Quantum Group Discrete Group Free Product Group Algebra 
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  1. 1.
    Connes, A.: Compact metric spaces, Fredholm modules, and hyperfiniteness. Ergodic Theor. Dyn. Syst. 9(2), 207–220 (1989)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Rieffel, M.A.: Metrics on states from actions of compact groups. Doc. Math. 3, 215–229 (1998)MATHMathSciNetGoogle Scholar
  3. 3.
    Rieffel, M.A.: Metrics on state spaces. Doc. Math. 4, 559–600 (1999)MATHMathSciNetGoogle Scholar
  4. 4.
    Rieffel, M.A.: Compact quantum metric spaces, operator algebras, quantization, and noncommutative geometry. Contemp. Math. 365, 315–330; Am. Math. Soc. Provid. RI (2004)Google Scholar
  5. 5.
    Christ, M., Rieffel, M.A.: Nilpotent group \(C^*\)-algebras as compact quantum metric spaces. arXiv:1508.00980
  6. 6.
    Ozawa, N., Rieffel, M.A.: Hyperbolic group \(C^*\)-algebras and free-product \(C^*\)-algebras as compact quantum metric spaces. Canad. J. Math. 57(5), 1056–1079 (2005)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Antonescu, C., Christensen, E.: Metrics on group \(C^*\)-algebras and a non-commutative Arzelà-Ascoli theorem. J. Funct. Anal. 214(2), 247–259 (2004)CrossRefMATHGoogle Scholar
  8. 8.
    Park, E.: Isometries of unbounded Fredholm modules over reduced group \(C^*\)-algebras. Proc. Am. Math. Soc. 123(6), 1839–1843 (1995)MATHMathSciNetGoogle Scholar
  9. 9.
    Bertozzini, P., Conti, R., Lewkeeratiyutkul, W.: A category of spectral triples and discrete groups with length function. Osaka J. Math. 43(2), 327–350 (2006)MATHMathSciNetGoogle Scholar
  10. 10.
    Bhowmick, J., Skalski, A.: Quantum isometry groups of noncommutative manifolds associated to group \(C^*\)-algebras. J. Geom. Phys. 60(10), 1474–1489 (2010)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Banica, T.: Quantum automorphism groups of homogeneous graphs. J. Funct. Anal. 224(2), 243–280 (2005)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Banica, T., Skalski, A.: Quantum isometry groups of duals of free powers of cyclic groups. Int. Math. Res. Not. 9, 2094–2122 (2012)MATHMathSciNetGoogle Scholar
  13. 13.
    Banica, T., Skalski, A.: Quantum symmetry groups of \(C^*\)-algebras equipped with orthogonal filtrations. Proc. Lond. Math. Soc. 106(3, 5), 980–1004 (2013)Google Scholar
  14. 14.
    Goswami, D., Mandal, A.: Quantum isometry groups of dual of finitely generated discrete groups and quantum groups. arXiv:1408.5683
  15. 15.
    Banica, T., Bichon, J., Collins, B.: Quantum permutation groups: a survey. In: Noncommutative Harmonic Analysis with Applications to Probability, pp. 13–34, vol. 78. Banach Center Publications. Polish Acad. Sci. Inst. Math., Warsaw (2007)Google Scholar
  16. 16.
    Banica, T., Skalski, A.: Two-parameter families of quantum symmetry groups. J. Funct. Anal. 260(11), 3252–3282 (2011)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Bichon, J.: Free wreath product by the quantum permutation group. Algebr. Represent. Theor. 7(4), 343–362 (2004)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Banica, T., Vergnioux, R.: Fusion rules for quantum reflection groups. J. Noncommut. Geom. 3(3), 327–359 (2009)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Mandal, A.: Quantum isometry group of dual of finitely generated discrete groups–II. Annales Mathematiques Blaise Pascal (2016). arXiv:1504.02240
  20. 20.
    Liszka-Dalecki, J., Sołtan, P.M.: Quantum isometry groups of symmetric groups. Internat. J. Math. 23(7) (2012)Google Scholar
  21. 21.
    Skalski, A.; Sołtan, P.M.: Projective limits of quantum symmetry groups and the doubling construction for Hopf algebras. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 17(2), 1450012 (27 pages) (2014)Google Scholar
  22. 22.
    Tao, J., Qiu, D.: Quantum isometry groups for Dihedral group \(D_{2(2n\,+\,1)}\). J. Geom. Phys. 62(9), 1977–1983Google Scholar
  23. 23.
    Maes, A., Van Daele, A.: Notes on compact quantum groups. Nieuw Arch. Wisk. 16(4, 1–2), 73–112 (1998)Google Scholar
  24. 24.
    Fukuda, N.: Semisimple Hopf algebras of dimension 12. Tsukuba. J. Math. 21, 43–54 (1997)MATHMathSciNetGoogle Scholar

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© Springer (India) Pvt. Ltd 2016

Authors and Affiliations

  1. 1.Statistics and Mathematics UnitIndian Statistical InstituteKolkataIndia

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