Weak amenability of the universal covering group of \(SU(1,n)\)
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 \(\widetilde{SU}(1,n)\).
connected, simple, real rank one Lie groups
scientific article; zbMATH DE number 4134777