Simple Lie groups without the approximation property

Abstract

For a locally compact group $G$, let $A\left(G\right)$ denote its Fourier algebra, and let $M_{0}A\left(G\right)$ denote the space of completely bounded Fourier multipliers on $G$. The group $G$ is said to have the Approximation Property (AP) if the constant function $1$ can be approximated by a net in $A\left(G\right)$ in the weak-∗ topology on the space $M_{0}A\left(G\right)$. Recently, Lafforgue and de la Salle proved that $SL(3,\mathbb{R})$ does not have the AP, implying the first example of an exact discrete group without it, namely, $SL(3,\mathbb{Z})$. In this paper we prove that $Sp(2,\mathbb{R})$ 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.