
Journal of Lie Theory 26 (2016), No. 3, 861887 Copyright Heldermann Verlag 2016 Group C*Algebras without the Completely Bounded Approximation Property Uffe Haagerup Dept. of Mathematics and Computer Science, University of Southern Denmark, Campusvej 55, 5230 Odense M, Denmark It is proved that: (1) The Fourier algebra A(G) of a simple Lie group G of real rank at least 2 with finite center does not have a multiplier bounded approximate unit. (2) The reduced C*algebra C_{r}*(Γ) of any lattice Γ in a noncompact simple Lie group of real rank at least 2 with finite center does not have the completely bounded approximation property. Hence, the results obtained by J. de Cannière and the author [Multipliers of the Fourier algebras of some simple Lie groups and their discrete subgroups, Amer. J. Math. 107 (1985) 455500] for SO_{e}(n,1), n ≥ 2, and by M. Cowling [Harmonic analysis on some nilpotent Lie groups (with application to the representation theory of some semisiple Lie groups), in: Topics in modern harmonic analysis, Vol. I, II (Turin/Milan, 1982), Ist. Naz. Alta Mat. Francesco Severi, Rome (1983) 81123] for SU(n,1) do not generalize to simple Lie groups of real rank at least 2. Keywords: Completely bounded approximation property, Group C*algebras, weak amenability, lattices in Lie groups. MSC: 43A22, 43A80, 22E40, 22D25, 22D15 [ Fulltextpdf (396 KB)] for subscribers only. 