For a locally compact group , let denote its Fourier algebra, and let denote the space of completely bounded Fourier multipliers on . The group is said to have the Approximation Property (AP) if the constant function can be approximated by a net in in the weak-∗ topology on the space . Recently, Lafforgue and de la Salle proved that does not have the AP, implying the first example of an exact discrete group without it, namely, . In this paper we prove that does not have the AP. It follows that all connected simple Lie groups with finite center and real rank greater than or equal to two do not have the AP. This naturally gives rise to many examples of exact discrete groups without the AP.
Uffe Haagerup. Tim de Laat. "Simple Lie groups without the approximation property." Duke Math. J. 162 (5) 925 - 964, 1 April 2013. https://doi.org/10.1215/00127094-2087672