Abstract

Motivated by a question of Vincent Lafforgue, we study the Banach spaces X satisfying the following property: there is a function ε → ∆X(ε) tending to zero with ε> 0 such that every operator T: L2 → L2 with ‖T ‖ ≤ ε that is simultaneously contractive (i.e. of norm ≤ 1) on L1 and on L ∞ must be of norm ≤ ∆X(ε) on L2(X). We show that ∆X(ε) ∈ O(ε α) for some α> 0 iff X is isomorphic to a quotient of a subspace of an ultraproduct of θ-Hilbertian spaces for some θ> 0 (see Corollary 6.7), where θ-Hilbertian is meant in a slightly more general sense than in our previous paper [43]. Let Br(L2(µ)) be the space of all regular operators on L2(µ). We are able to describe the complex interpolation space (Br(L2(µ)),B(L2(µ))) θ. We show that T: L2(µ) → L2(µ) belongs to this space iff T ⊗ idX is bounded on L2(X) for any θ-Hilbertian space X. More generally, we are able to describe the spaces (B(ℓp0),B(ℓp1))θ or (B(Lp0),B(Lp1))θ for any pair 1 ≤ p0,p1 ≤ ∞ and 0 < θ < 1. In the same vein, given a locally compact Abelian group G, let M(G) (resp. PM(G)) be the space of complex measures (resp. pseudo-measures) on

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