Approximation properties for noncommutative Lp-spaces of high rank lattices and nonembeddability of expanders.

This article contains two rigidity type results for SL(n,Z) for large n that share the same proof. Firstly, we prove that for every p ∈ [1,∞] different from 2, the noncommutative Lp-space associated with SL(n,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 SL(n,Z). Let X be a Banach space and suppose that there exist β < 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 Ckβ. Then the expander family constructed from SL(n,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. [ABSTRACT FROM AUTHOR]