Approximation properties for noncommutative Lp-spaces associated with lattices in Lie groups

In 2010, Lafforgue and de la Salle gave examples of noncommutative L^p-spaces without the operator space approximation property (OAP) and, hence, without the completely bounded approximation property (CBAP). To this purpose, they introduced the property of completely bounded approximation by Schur multipliers on S^p and proved that for p in [1,4/3) or in (4,infinity], the groups SL(n,Z), with n > 2, do not have this property. Since for p in (1, infinity), the property of completely bounded approximation by Schur multipliers on S^p is weaker than the approximation property of Haagerup and Kraus (AP), these groups were also the first examples of exact groups without the AP. Recently, Haagerup and the author proved that also the group Sp(2,R) does not have the AP, without using the property of completely bounded approximation by Schur multipliers on S^p. In this paper, we prove that Sp(2,R) does not have the property of completely bounded approximation by Schur multipliers on S^p for p in [1,12/11) or in (12,infinity]. It follows that a large class of noncommutative L^p-spaces does not have the OAP or CBAP.