Weak amenability of the universal covering group of \(SU(1,n)\)
type: Item
instance of: scholarly article
title: Weak amenability of the universal covering group of \(SU(1,n)\)
zbMATH Open document ID: 0693.43004
DOI: 10.1007/BF01444541
author: Mogens Lemvig Hansen
published in: Mathematische Annalen
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164743
review text: By a recent result of \textit{M. Cowling} and \textit{U. Haagerup} [Invent. Math. 96, 507-549 (1989; Zbl 0681.43012)] every connected, simple Lie group \(G\) of real rank one with finite center is weakly amenable in the sense, that for some \(L\geq 1\) there exists a net \(\psi_{\alpha}\) in the Fourier algebra \(A(G)\) of \(G\), such that \(\| \psi_{\alpha}\|_{M_ 0A(G)}\leq L\), and \(\psi_{\alpha}\to 1\) uniformly on compacta. We extend this result to all connected, simple, real rank one Lie groups by proving, that for \(n\geq 1\) the universal covering group of \(SU(1,n)\) is weakly amenable with constant \(L=1.\) As in the above cited paper, the proof relies on a computation of upper bounds of \(M_ 0 A(G)\)-norms on a class of well-behaved functions on \(G\). In the case \(G=\widetilde{SU}(1,n)\) we can use \textit{M. Flensted-Jensen}'s spherical functions \(\phi_{\lambda,\mu}\), \(\lambda\), \(\mu\in {\mathbb{C}}\) [Math. Ann. 228, 65-92 (1977; Zbl 0331.43013)], and the upper bounds are computed by realizing these spherical functions as coefficients of concretely described representations of \(\widetilde{SU}(1,n)\).
Mathematics Subject Classification ID: 43A15
Mathematics Subject Classification ID: 43A90
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 43A07
Mathematics Subject Classification ID: 43A22
Mathematics Subject Classification ID: 43A30
Mathematics Subject Classification ID: 22D25
zbMATH DE Number: 4134777
zbMATH Keywords: weakly amenable
zbMATH Keywords: Fourier algebra
zbMATH Keywords: connected, simple, real rank one Lie groups
zbMATH Keywords: universal covering group
zbMATH Keywords: spherical functions
zbMATH Keywords: representations
MaRDI profile type: Publication
cites work: Q5558293
cites work: Q4112938
cites work: Completely bounded multipliers of the Fourier algebra of a simple Lie group of real rank one
cites work: Multipliers of the Fourier Algebras of Some Simple Lie Groups and Their Discrete Subgroups
cites work: Spherical Functions on a Simply Connected Semisimple Lie Group
cites work: Q3671491
description: scientific article; zbMATH DE number 4134777