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Journal of Lie Theory 26 (2016), No. 3, 861--887
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 Cr*(Γ) of any lattice Γ in a non-compact 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) 455--500] for SOe(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) 81--123] 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

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