### Fourier multiplier norms of spherical functions on the generalized Lorentz groups

Troels Steenstrup

#### Abstract

Our main result provides a closed expression for the completely bounded Fourier multiplier norm of the spherical functions on the generalized Lorentz groups $SO_0(1,n)$ (for $n \geq 2$). As a corollary, we find that there is no uniform bound on the completely bounded Fourier multiplier norm of the spherical functions on the generalized Lorentz groups. We extend the latter result to the groups $SU(1,n)$, $Sp(1,n)$ (for $n \geq 2$) and the exceptional group $F_{4(-20)}$, and as an application we obtain that each of the above mentioned groups has a completely bounded Fourier multiplier, which is not the coefficient of a uniformly bounded representation of the group on a Hilbert space.

#### Article information

Source
Adv. Oper. Theory, Volume 3, Number 1 (2018), 193-230.

Dates
Accepted: 19 June 2017
First available in Project Euclid: 5 December 2017

https://projecteuclid.org/euclid.aot/1512497959

Digital Object Identifier
doi:10.22034/aot.1706-1172

Mathematical Reviews number (MathSciNet)
MR3730346

Zentralblatt MATH identifier
1379.43009

#### Citation

Steenstrup, Troels. Fourier multiplier norms of spherical functions on the generalized Lorentz groups. Adv. Oper. Theory 3 (2018), no. 1, 193--230. doi:10.22034/aot.1706-1172. https://projecteuclid.org/euclid.aot/1512497959

#### References

• M. Abramowitz and I. A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, 55 For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 1964.
• M. Bożejko and G. Fendler, Herz–Schur multipliers and completely bounded multipliers of the Fourier algebra of a locally compact group, Boll. Un. Mat. Ital. A (6) 3 (1984), no. 2, 297–302.
• M. Bożejko and G. Fendler, Herz–Schur multipliers and uniformly bounded representations of discrete group, Arch. Math. (Basel) 57 (1991), no. 3, 290–298.
• P.-A. Cherix, M. Cowling, P. Jolissaint, P. Julg, and A. Valette, Groups with the Haagerup property. Gromov's a-T-menability, volume 197 of Progress in Mathematics, Birkhäuser Verlag, Basel, 2001.
• J.-L. Clerc, P. Eymard, J. Faraut, M. Rais, and R. Takahashi, Analyse harmonique, Les cours du C. I. M. P. A., Nice, 1983.
• M. Cowling and U. Haagerup, Completely bounded multipliers of the Fourier algebra of a simple Lie group of real rank one, Invent. Math. 96 (1989), no. 3, 507–549.
• J. De Cannière and U. Haagerup, Multipliers of the Fourier algebras of some simple Lie groups and their discrete subgroups, Amer. J. Math. 107 (1985), no. 2, 455–500.
• J.-L. Dunau, Etude d'une classe de marches alèatoires sur l'arbre homogéne, Publications du Laboratoire de Statistique et Probabilités, (4), 1976.
• A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher transcendental functions. Vol. I, McGraw-Hill Book Company, Inc., New York–Toronto–London, 1953.
• A.Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher transcendental functions. Vol. II. McGraw-Hill Book Company, Inc., New York–Toronto–London, 1953.
• P. Eymard, L'algèbre de Fourier d'un groupe localement compact, Bull. Soc. Math. France 92 1964 181–236.
• A. Figà-Talamanca and C. Nebbia, Harmonic analysis and representation theory for groups acting on homogeneous trees, volume 162 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1991.
• A. Figà-Talamanca and M. A. Picardello, Harmonic analysis on free groups, volume 87 of Lecture Notes in Pure and Applied Mathematics, Marcel Dekker Inc., New York, 1983.
• F. P. Greenleaf, Invariant means on topological groups and their applications, Van Nostrand Mathematical Studies, No. 16. Van Nostrand Reinhold Co., New York, 1969.
• R. Gangolli and V. S. Varadarajan, Harmonic analysis of spherical functions on real reductive groups, volume 101 of Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], Springer-Verlag, Berlin, 1988.
• Harish-Chandra, Spherical functions on a semisimple Lie group. I, Amer. J. Math. 80 (1958), 241–310.
• S. Helgason, Differential geometry, Lie groups, and symmetric spaces, volume 80 of Pure and Applied Mathematics, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1978.
• S. Helgason, Groups and geometric analysis. Integral geometry, invariant differential operators, and spherical functions, volume 113 of Pure and Applied Mathematics, Academic Press Inc., Orlando, FL, 1984.
• C. Herz, Une généralisation de la notion de transformée de Fourier-Stieltjes (French) Ann. Inst. Fourier (Grenoble) 24 (1974), no. 3, xiii, 145–157.
• U. Haagerup, T. Steenstrup, and R. Szwarc, Schur multipliers and spherical functions on homogeneous trees, Internat. J. Math. 21 (2010), no. 10, 1337–1382.
• P. Jolissaint, A characterization of completely bounded multipliers of Fourier algebras, Colloq. Math. 63 (1992), no. 2, 311–313.
• S. Knudby, The weak Haagerup property, Trans. Amer. Math. Soc. 368 (2016), no. 5, 3469–3508.
• R. L. Lipsman, Uniformly bounded representations of the Lorentz groups, Amer. J. Math. 91 (1969), 938–962.
• G. Pisier, Are unitarizable groups amenable?, Infinite groups: geometric, combinatorial and dynamical aspects, 323–362, Progr. Math., 248, Birkhäuser, Basel, 2005.
• W. Rudin, Functional analysis, International Series in Pure and Applied Mathematics. McGraw-Hill Inc., New York, second edition, 1991.
• R. L. Schilling, Measures, integrals and martingales, Cambridge University Press, New York, 2005.
• T. Steenstrup, Herz–Schur multipliers and non-uniformly bounded representations of locally compact groups, Probab. Math. Statist. 33 (2013), no. 2, 213–223.
• R. Takahashi, Sur les représentations unitaires des groupes de Lorentz généralisés, Bull. Soc. Math. France 91 (1963), 289–433.