Advances in Operator Theory

Fourier multiplier norms of spherical functions on the generalized Lorentz groups

Troels Steenstrup

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

Permanent link to this document
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

Subjects
Primary: 22E46: Semisimple Lie groups and their representations
Secondary: 43A90: Spherical functions [See also 22E45, 22E46, 33C55] 46L07: Operator spaces and completely bounded maps [See also 47L25]

Keywords
Lie group completely bounded Fourier multiplier norm generalized Lorentz group representation spherical function

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


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