Group \(C^{*}\)-algebras without the completely bounded approximation property (Q2827856)

From Søren Knudby's introduction:NEWLINENEWLINEThe Fourier algebra \(A(G)\) of a locally compact group, introduced by \textit{P. Eymard} [Bull. Soc. Math. Fr. 92, 181--236 (1964; Zbl 0169.46403)], consists of the matrix coefficients of the regular representation. The Fourier algebra is the predual of the group von Neumann algebra \(\mathfrak{M}(G)\) generated by the regular representation.NEWLINENEWLINE\textit{H. Leptin} [C. R. Acad. Sci., Paris, Sér. A 266, 1180--1182 (1968; Zbl 0169.46501)] showed that the Fourier algebra \(A(G)\) has an approximate unit bounded in norm if and only if \(G\) is amenable. In [Am. J. Math. 107, 455--500 (1985; Zbl 0577.43002)], \textit{J. de Cannière} and the author showed that the Fourier algebra of the non-amenable group \(\mathrm{SO}_e(n,1)\), \(n\geq 2\), admits an approximate unit bounded in the completely bounded multiplier norm. In [Proc. Semin., Torino and Milano 1982, Vol. 1, 81--123 (1983; Zbl 0541.22005)], \textit{M. G. Cowling} obtained similar results for \(\mathrm{SU}(n,1)\).NEWLINENEWLINEIn the first half of the present paper the author shows that these results do not generalize to simple Lie groups of real rank at least 2:NEWLINENEWLINETheorem 1. The Fourier algebra \(A(G)\) of a simple Lie group \(G\) of real rank at least 2 with finite center does not have an approximate unit bounded in multiplier norm.NEWLINENEWLINEThe second half of the paper is concerned with applications to operator algebras. It is shown that the Fourier algebra \(A(\Gamma)\) of a lattice \(\Gamma\) in a second countable locally compact group \(G\) has an approximate unit bounded in the completely bounded multiplier norm if and only if the Fourier algebra \(A(G)\) of \(G\) has such an approximate unit (Theorem 2.3).