Abstract. Consider a completely bounded Fourier multiplier φ of a locally compact group G, and take 1 ≤ p ≤ ∞. One can associate to φ a Schur multiplier on the Schatten classes Sp(L 2 G), as well as a Fourier multiplier on L p (LG), the non-commutative L p -space of the group von Neumann algebra of G. We prove that the completely bounded norm of the Schur multiplier is not greater than the completely bounded norm of the L p -Fourier multiplier. When G is amenable we show that equality holds, extending a result by Neuwirth and Ricard to non-discrete groups.For a discrete group G and in the special case when p = 2 is an even integer, we show the following. If there exists a map between L p (LG) and an ultraproduct of L p (M) ⊗ Sp(L 2 G) that intertwines the Fourier multiplier with the Schur multiplier, then G must be amenable. This is an obstruction to extend the Neuwirth-Ricard result to non-amenable groups.

Abstract. 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 L p -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 Ck β . 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. We prove that connected higher rank simple Lie groups have Lafforgue's strong property (T) with respect to a certain class of Banach spaces E 10 containing many classical superreflexive spaces and some non-reflexive spaces as well. This generalizes the result of Lafforgue asserting that SL(3, R) has strong property (T) with respect to Hilbert spaces and the more recent result of the second named author asserting that SL(3, R) has strong property (T) with respect to a certain larger class of Banach spaces. For the generalization to higher rank groups, it is sufficient to prove strong property (T) for Sp(2, R) and its universal covering group. As consequences of our main result, it follows that for X ∈ E 10 , connected higher rank simple Lie groups and their lattices have property (F X ) of Bader, Furman, Gelander and Monod, and that the expanders contructed from a lattice in a connected higher rank simple Lie group do not admit a coarse embedding into X.