Approximation properties for noncommutative $L^p$-spaces of high rank   lattices and nonembeddability of expanders
  This article contains two rigidity type results for $\mathrm{SL}(n,\mathbb{Z})$ for large $n$ that share the same proof. Firstly, we prove that for every $p \in [1,\infty]$ different from $2$, the noncommutative $L^p$-space associated with $\mathrm{SL}(n,\mathbb{Z})$ does not have the completely bounded approximation property for sufficiently large $n$ depending on $p$.   The second result concerns the coarse embeddability of expander families constructed from $\mathrm{SL}(n,\mathbb{Z})$. Let $X$ be a Banach space and suppose that there exist $\beta < \frac{1}{2}$ and $C>0$ such that the Banach-Mazur distance to a Hilbert space of all $k$-dimensional subspaces of $X$ is bounded above by $C k^\beta$. Then the expander family constructed from $\mathrm{SL}(n,\mathbb{Z})$ does not coarsely embed into $X$ for sufficiently large $n$ depending on $X$.   More generally, we prove that both results hold for lattices in connected simple real Lie groups with sufficiently high real rank. 
Tim de Laat and Mikael de la Salle
math.OA math.FA math.GR math.MG