Simple Lie groups without the Approximation Property

TL;DR

This paper demonstrates that all connected simple Lie groups with finite center and real rank at least two lack the Approximation Property, providing new examples of exact discrete groups without this property.

Contribution

It proves that Sp(2,R) and all similar higher-rank simple Lie groups do not have the Approximation Property, extending previous results to a broader class of groups.

Findings

Sp(2,R) does not have the AP.

All connected simple Lie groups with real rank ≥ 2 lack the AP.

Many exact discrete groups without the AP are identified.

Abstract

For a locally compact group G, let A(G) denote its Fourier algebra, and let M_0A(G) 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(G) in the weak-* topology on the space M_0A(G). Recently, Lafforgue and de la Salle proved that SL(3,R) does not have the AP, implying the first example of an exact discrete group without it, namely SL(3,Z). In this paper we prove that Sp(2,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.