Simple Lie groups without the approximation property. II (Q2790608)

The central result shown in the paper at hand is the fact that the universal covering group \(\widetilde{\mathrm{Sp}(2,\mathbb{R})}\) of \(\mathrm{Sp}(2,\mathbb{R})\) does not have the Approximation Property (AP). Together with the fact that \(\mathrm{SL}(3,\mathbb{R})\) does not have the AP, which was proved by \textit{V. Lafforgue} and \textit{M. De La Salle} [Duke Math. J. 160, No. 1, 71--116 (2011; Zbl 1267.46072)], and the fact that \(\mathrm{Sp}(2,\mathbb{R})\) does not have the AP, which was proved by the authors in Part I of the paper [Duke Math. J. 162, No. 5, 925--964 (2013; Zbl 1266.22008)], this completes the description of the AP for connected simple Lie groups. Indeed, it follows that a connected simple Lie group has the AP if and only if its real rank is zero or one.NEWLINENEWLINEBy an adaptation of the methods they use to study the AP, the authors obtain results on approximation properties for noncommutative \(L^p\)-spaces associated with lattices in \(\widetilde{\mathrm{Sp}(2,\mathbb{R})}\). Combining this with earlier results of Lafforgue and la Salle [loc. cit.] and results of U. Haagerup, this gives rise to results on approximation properties of noncommutative \(L^p\)-spaces associated with lattices in any connected simple Lie group.