Journal of Fourier Analysis and Applications

, Volume 15, Issue 3, pp 336–365

On Twisted Fourier Analysis and Convergence of Fourier Series on Discrete Groups

Article

Abstract

We study norm convergence and summability of Fourier series in the setting of reduced twisted group C*-algebras of discrete groups. For amenable groups, Følner nets give the key to Fejér summation. We show that Abel-Poisson summation holds for a large class of groups, including e.g. all Coxeter groups and all Gromov hyperbolic groups. As a tool in our presentation, we introduce notions of polynomial and subexponential H-growth for countable groups w.r.t. proper scale functions, usually chosen as length functions. These coincide with the classical notions of growth in the case of amenable groups.

Keywords

Twisted group C*-algebra Fourier series Fejér summation Abel-Poisson summation Amenable group Haagerup property Length function Polynomial growth Subexponential growth 

Mathematics Subject Classification (2000)

22D10 22D25 46L55 43A07 43A65 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Akemann, C.A., Ostrand, P.A.: Computing norms in group C *-algebras. Am. J. Math. 98, 1015–1047 (1976) MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Backhouse, N.B.: Projective representations of space groups, II: Factor systems. J. Math. Oxford 21, 223–242 (1970) CrossRefGoogle Scholar
  3. 3.
    Backhouse, N.B., Bradley, C.J.: Projective representations of space groups, I: Translation groups. J. Math. Oxford 21, 203–222 (1970) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bédos, E., Conti, R.: On infinite tensor products of projective unitary representations. Rocky Mt. J. Math. 34, 467–493 (2004) MATHCrossRefGoogle Scholar
  5. 5.
    Bédos, E., Conti, R.: Fourier series and twisted C*-crossed products. In preparation Google Scholar
  6. 6.
    Bellissard, J.: Gap labelling theorems for Schrödinger operators. In: From number theory to physics, Les Houches, 1989, pp. 538–630. Springer, Berlin (1992) Google Scholar
  7. 7.
    Bellissard, J.: The noncommutative geometry of aperiodic solids. In: Geometric and topological methods for quantum field theory, Villa de Leya, 2001, pp. 86–156. World Scientific, Singapore (2003) CrossRefGoogle Scholar
  8. 8.
    Berg, C., Reus Christensen, J.P., Ressel, P.: Harmonic Analysis on Semigroups. GTM, vol. 100. Springer, Berlin (1984) MATHGoogle Scholar
  9. 9.
    Bożejko, M.: Positive definite bounded matrices and a characterisation of amenable groups. Proc. Am. Math. Soc. 95, 357–360 (1985) MATHCrossRefGoogle Scholar
  10. 10.
    Bożejko, M.: Positive-definite kernels, length functions on groups and a noncommutative von Neumann inequality. Stud. Math. XCV, 107–118 (1989) Google Scholar
  11. 11.
    Bożejko, M., Fendler, G.: Herz-Schur multipliers and completely bounded multipliers of the Fourier algebra of a locally compact group. Boll. Unione Mat. Ital. A 3(6), 297–302 (1984) MATHGoogle Scholar
  12. 12.
    Bożejko, M., Januszkiewicz, T., Spatzier, R.J.: Infinite Coxeter groups do not have Kazhdan’s property. J. Oper. Theory 19, 63–67 (1988) MATHGoogle Scholar
  13. 13.
    Bożejko, M., Picardello, M.A.: Weakly amenable groups and amalgameted products. Proc. Am. Math. Soc. 117, 1039–1046 (1993) MATHCrossRefGoogle Scholar
  14. 14.
    Brodzki, J., Niblo, G.: Approximation properties for discrete groups. In: C *-algebras and elliptic theory. Trends in Mathematics, pp. 23–35. Birkhäuser, Basel (2006) CrossRefGoogle Scholar
  15. 15.
    Carey, A.L., Hannabuss, K.A., Mathai, V., McCann, P.: Quantum Hall effect on the hyperbolic plane. Commun. Math. Phys. 190, 629–673 (1998) MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Chatterji, I.: Twisted rapid decay. Appendix to [65] Google Scholar
  17. 17.
    Chatterji, I., Ruane, K.: Some geometric groups with rapid decay. Geom. Funct. Anal. 15, 311–339 (2005) MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Chen, X., Wei, S.: Spectral invariant subalgebras of reduced crossed product C *-algebras. J. Funct. Anal. 197, 228–246 (2003) MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Cherix, P.-A., Cowling, M., Jolissaint, P., Julg, P., Valette, A.: Groups with the Haagerup property. In: Gromov’s a-T-menability. Progress in Mathematics, vol. 197. Birkhäuser, Basel (2001) Google Scholar
  20. 20.
    Cohen, J.M.: Operator norms in free groups. Boll. Unione Mat. Ital., B 1, 1055–1065 (2003) Google Scholar
  21. 21.
    Connes, A.: Compact metric spaces, Fredholm modules, and hyperfiniteness. Ergod. Theory Dyn. Syst. 9, 207–220 (1989) MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Connes, A.: Noncommutative Geometry. Academic Press, New York (1994) MATHGoogle Scholar
  23. 23.
    Cowling, M.: Sur les coefficients des représentations des groupes Lie simples. Lect. Not. Math. 739, 132–178 (1979) CrossRefMathSciNetGoogle Scholar
  24. 24.
    Cowling, M.: Harmonic analysis on some nilpotent Lie groups (with applications to the representation theory of some semisimple Lie groups). In: Topics in modern harmonic analysis, Turin/Milan, 1982. Ist. Naz.Alta Mat., vols. I, II, pp. 81–123. Francesco Severi, Rome (1983) Google Scholar
  25. 25.
    Cowling, M., Haagerup, U.: Completely bounded multipliers of the Fourier algebra of a simple Lie group of real rank one. Invent. Math. 96, 507–549 (1989) MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Davidson, K.R.: C *-Algebras by Examples. Fields Institute Monographs, vol. 6. Am. Math. Soc., Providence (1996) Google Scholar
  27. 27.
    de Cannière, J., Haagerup, U.: Multipliers of the Fourier algebras of some simple Lie groups and their discrete subgroups. Am. J. Math. 107, 455–500 (1985) MATHCrossRefGoogle Scholar
  28. 28.
    de la Harpe, P.: Groupes de Coxeter infinis non affines. Exp. Math. 5, 91–96 (1987) MATHGoogle Scholar
  29. 29.
    de la Harpe, P.: Groupes hyperboliques, algèbres d’ opérateurs et un théorème de Jolissaint. C. R. Acad. Sci. Paris, Sér. I 307, 771–774 (1988) MATHGoogle Scholar
  30. 30.
    de la Harpe, P.: Topics in Geometric Group Theory. Chicago Lectures in Mathematics Series. University of Chicago Press, Chicago (2000) MATHGoogle Scholar
  31. 31.
    Dixmier, J.: Les C *-Algèbres et Leurs Représentations. Gauthiers-Villars, Paris (1969) Google Scholar
  32. 32.
    Dixmier, J.: Les Algèbres d’Opérateurs dans l’Espace Hilbertien (Algèbres de von Neumann). Gauthiers-Villars, Paris (1969) MATHGoogle Scholar
  33. 33.
    Dixmier, J.: Topologie Générale. PUF, Paris (1981) MATHGoogle Scholar
  34. 34.
    Dorofaeff, B.: The Fourier algebra of SL(2,ℝ)n,n≥2, has no multiplier unit. Math. Ann. 297, 707–724 (1993) MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Effros, E.G., Ruan, Z.-J.: Multivariable multiplier for groups and their operator algebras. In: Operator Theory: Operator Algebras and Their Applications, Part I, Durham, NH, 1988. Proc. Symp. Pure Math., vol. 51, pp. 197–218. Am. Math. Soc., Providence (1990) Google Scholar
  36. 36.
    Exel, R.: Hankel matrices over right ordered amenable groups. Can. Math. Bull. 33, 404–415 (1990) MATHMathSciNetGoogle Scholar
  37. 37.
    Eymard, P.: L’algèbre de Fourier d’un groupe localement compact. Bull. Soc. Math. France 92, 181–236 (1964) MATHMathSciNetGoogle Scholar
  38. 38.
    Fendler, G.: Simplicity of the reduced C *-algebras of certain Coxeter groups. Illinois J. Math. 47, 883–897 (2003) MATHMathSciNetGoogle Scholar
  39. 39.
    Fendler, G., Gröchenig, K., Leinert, M.: Symmetry of weighted L 1-algebras and the GRS-condition. Bull. Lond. Math. Soc. 38, 625–635 (2006) MATHCrossRefGoogle Scholar
  40. 40.
    Figà-Talamanca, A., Picardello, M.A.: Harmonic Analysis on Free Groups. Lectures Notes in Pure and Appl. Math., vol. 87. Dekker, New York (1983) MATHGoogle Scholar
  41. 41.
    Flory, V.: Estimating norms in C *-algebras of discrete groups. Math. Ann. 224, 41–52 (1976) MATHCrossRefMathSciNetGoogle Scholar
  42. 42.
    Ghys, E., de la Harpe, P.: Sur les Groupes Hyperboliques d’Après Mikhael Gromov. Progress in Math., vol. 83. Birkhaüser, Basel (1990) MATHGoogle Scholar
  43. 43.
    Gröchenig, K., Leinert, M.: Wiener’s lemma for twisted convolution and Gabor frames. J. Am. Math. Soc. 17, 1–18 (2004) MATHCrossRefGoogle Scholar
  44. 44.
    Gromov, M.: Hyperbolic groups. Math. Sci. Res. Inst. Publ. 8, 75–263 (1987) MathSciNetGoogle Scholar
  45. 45.
    Haagerup, U.: An example of a nonnuclear C *-algebra, which has the metric approximation property. Invent. Math. 50, 279–293 (1978/79) CrossRefMathSciNetGoogle Scholar
  46. 46.
    Haagerup, U.: Group C *-algebras without the completely bounded approximation property. Unpublished manuscript (1986) Google Scholar
  47. 47.
    Haagerup, U., Kraus, J.: Approximation properties for group C *-algebras and group von Neumann algebras. Trans. Am. Math. Soc. 344, 667–699 (1994) MATHCrossRefMathSciNetGoogle Scholar
  48. 48.
    Higson, N., Guentner, E.: Group C *-algebras and K-theory. In: Noncommutative geometry. Lecture Notes in Math., vol. 1831, pp. 137–251. Springer, Berlin (2004) Google Scholar
  49. 49.
    Higson, N., Guentner, E.: Weak amenability of CAT(0)-cubical groups. Preprint (2007) Google Scholar
  50. 50.
    Howlett, R.B.: On the Schur multipliers of Coxeter groups. J. Lond. Math. Soc. 38, 263–276 (1988) MATHCrossRefMathSciNetGoogle Scholar
  51. 51.
    Humphreys, J.E.: Reflection Groups and Coxeter Groups. Cambridge University Press, Cambridge (1990) MATHGoogle Scholar
  52. 52.
    Januszkiewicz, T.: For Coxeter groups z |g| is a coefficient of a uniformly bounded representation. Fund. Math. 174, 79–86 (2002) MATHCrossRefMathSciNetGoogle Scholar
  53. 53.
    Ji, R., Schweitzer, L.: Spectral invariance of smooth crossed products, and rapid decay locally compact groups. K-Theory 10, 283–305 (1996) MATHCrossRefMathSciNetGoogle Scholar
  54. 54.
    Jolissaint, P.: Rapidly decreasing functions in reduced C *-algebras of groups. Trans. Am. Math. Soc. 317, 167–196 (1990) MATHCrossRefMathSciNetGoogle Scholar
  55. 55.
    Jolissaint, P.: K-theory of reduced C *-algebras and rapidly decreasing functions on groups. K-Theory 2, 167–196 (1990) Google Scholar
  56. 56.
    Jolissaint, P., Valette, A.: Normes de Sobolev et convoluteurs bornés sur L 2(G). Ann. Inst. Fourier (Grenoble) 41, 797–822 (1991) MathSciNetGoogle Scholar
  57. 57.
    Kleppner, A.: The structure of some induced representations. Duke Math. J. 29, 555–572 (1962) MATHCrossRefMathSciNetGoogle Scholar
  58. 58.
    Kleppner, A.: Multipliers on Abelian groups. Math. Ann. 158, 11–34 (1965) MATHCrossRefMathSciNetGoogle Scholar
  59. 59.
    Leinert, M.: Faltungsoperatoren auf gewissen diskreten Gruppen. Stud. Math. LII, 149–158 (1974) MathSciNetGoogle Scholar
  60. 60.
    Luef, F.: Gabor analysis, noncommutative tori and Feichtinger’s algebra. In: Gabor and Wavelet Frames. IMS Lecture Notes Series, vol. 10. World Scientific, Singapore (2005). arXiv:math/0504146v1 Google Scholar
  61. 61.
    Luef, F.: On spectral invariance of non-commutative tori. In: Operator Theory, Operator Algebras, and Applications. Contemp. Math., vol. 414, pp. 131–146. Am. Math. Soc., Providence (2006) Google Scholar
  62. 62.
    Mackey, G.: Unitary representations of group extensions I. Acta Math. 99, 265–311 (1958) MATHCrossRefMathSciNetGoogle Scholar
  63. 63.
    Marcolli, M., Mathai, V.: Twisted index theory on good orbifolds, I: noncommutative Bloch theory. Commun. Contemp. Math. 1, 553–587 (1999) MATHCrossRefMathSciNetGoogle Scholar
  64. 64.
    Marcolli, M., Mathai, V.: Towards the fractional quantum Hall effect: a noncommutative geometry perspective. Preprint (2005) Google Scholar
  65. 65.
    Mathai, V.: Heat kernels and the range of the trace on completions of twisted group algebras. With an appendix by Indira Chatterji. Contemp. Math. 398, 321–345 (2006) MathSciNetGoogle Scholar
  66. 66.
    Mercer, R.: Convergence of Fourier series in discrete crossed products of von Neumann algebras. Proc. Am. Math. Soc. 94, 254–258 (1985) MATHCrossRefMathSciNetGoogle Scholar
  67. 67.
    Nebbia, C.: Multipliers and asymptotic behaviour of the Fourier algebra of nonamenable groups. Proc. Am. Math. Soc. 84, 549–554 (1982) MATHCrossRefMathSciNetGoogle Scholar
  68. 68.
    Ol’shanskiĭ, A.Yu.: Almost every group is hyperbolic. Int. J. Algebra Comput. 2, 1–17 (1992) CrossRefGoogle Scholar
  69. 69.
    Ozawa, N.: Weak amenability of hyperbolic groups. Preprint (2007) Google Scholar
  70. 70.
    Ozawa, N., Rieffel, M.: Hyperbolic group C *-algebras and free-product C *-algebras as compact quantum metric spaces. Can. J. Math. 57, 1056–1079 (2005) MATHMathSciNetGoogle Scholar
  71. 71.
    Packer, J.A.: C *-algebras generated by projective representations of the discrete Heisenberg group. J. Oper. Theory 18, 41–66 (1987) MATHMathSciNetGoogle Scholar
  72. 72.
    Packer, J.A.: Twisted group C *-algebras corresponding to nilpotent discrete groups. Math. Scand. 64, 109–122 (1989) MATHMathSciNetGoogle Scholar
  73. 73.
    Packer, J.A., Raeburn, I.: Twisted crossed product of C *-algebras. Math. Proc. Camb. Philos. Soc. 106, 293–311 (1989) MATHMathSciNetGoogle Scholar
  74. 74.
    Packer, J.A., Raeburn, I.: Twisted crossed product of C *-algebras II. Math. Ann. 287, 595–612 (1990) MATHCrossRefMathSciNetGoogle Scholar
  75. 75.
    Packer, J.A., Raeburn, I.: On the structure of twisted group C *-algebras. Trans. Am. Math. Soc. 334, 685–718 (1992) MATHCrossRefMathSciNetGoogle Scholar
  76. 76.
    Paterson, A.: Amenability. Math. Surveys and Monographs, vol. 29. Am. Math. Soc., Providence (1988) MATHGoogle Scholar
  77. 77.
    Paulsen, V.: Completely Bounded Maps and Operator Algebras. Cambridge University Press, Cambridge (2002) MATHGoogle Scholar
  78. 78.
    Picardello, M.A.: Positive definite functions and Lp convolutions operators on amalgams. Pac. J. Math. 123, 209–221 (1986) MATHMathSciNetGoogle Scholar
  79. 79.
    Pier, J.P.: Amenable Locally Compact Groups. Wiley, New York (1984) Google Scholar
  80. 80.
    Pisier, G.: Similarity Problems and Completely Bounded Maps, 2nd edn. Lect. Notes in Math., vol. 1618. Springer, Berlin (2001) MATHGoogle Scholar
  81. 81.
    Sauvageot, J.-L.: Strong Feller noncommutative kernels, strong Feller semigroups and harmonic analysis. In: Operator Algebras and Quantum Field Theory, Rome, 1996, pp. 105–110. International Press, Cambridge (1997) Google Scholar
  82. 82.
    Sauvageot, J.-L.: Strong Feller semigroups on C *-algebras. J. Oper. Theory 42, 83–102 (1999) MATHMathSciNetGoogle Scholar
  83. 83.
    Schweitzer, L.B.: Dense m-convex Fréchet subalgebras of operator algebra crossed products by Lie groups. Int. J. Math. 4, 601–673 (1993) MATHCrossRefMathSciNetGoogle Scholar
  84. 84.
    Valette, A.: Les représentations uniformement bornées associées à un arbre réel. Bull. Soc. Math. Belg., Ser. A 42, 747–760 (1990) MATHMathSciNetGoogle Scholar
  85. 85.
    Valette, A.: Weak amenability of right-angled Coxeter groups. Proc. Am. Math. Soc. 119, 1331–1334 (1993) MATHCrossRefMathSciNetGoogle Scholar
  86. 86.
    Valette, A.: Introduction to the Baum-Connes Conjecture. Lectures in Mathematics ETH Zürich. Birkhäuser, Basel (2002) MATHGoogle Scholar
  87. 87.
    Wagon, S.: The Banach-Tarski Paradox. Encyclopedia of Math. and its Appl., vol. 24. Cambridge University Press, Cambridge (1985) MATHGoogle Scholar
  88. 88.
    Weaver, N.: Lipschitz algebras and derivations of von Neumann algebras. J. Funct. Anal. 139, 261–300 (1996) MATHCrossRefMathSciNetGoogle Scholar
  89. 89.
    Zeller-Meier, G.: Produits croisés d’une C *-algèbre par un groupe d’automorphismes. J. Math. Pures Appl. 47, 101–239 (1968) MATHMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Boston 2009

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of OsloOsloNorway
  2. 2.Mathematics, School of Mathematical and Physical SciencesUniversity of NewcastleCallaghanAustralia
  3. 3.Department of MathematicsUniversity of Rome 2 Tor VergataRomeItaly

Personalised recommendations