Complex Interpolation between Hilbert, Banach and Operator Spaces

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 : L 2 → L 2 with T ≤ ε that is simultaneously contractive (i.e. of norm ≤ 1) on L 1 and on L ∞ must be of norm ≤ ∆ X (ε) on L 2 (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 B r (L 2 (µ)) be the space of all regular operators on L 2 (µ). We are able to describe the complex interpolation space (B r (L 2 (µ)), B(L 2 (µ))) θ. We show that T : L 2 (µ) → L 2 (µ) belongs to this space iff T ⊗ id X is bounded on L 2 (X) for any θ-Hilbertian space X. More generally, we are able to describe the spaces (B(ℓ p 0), B(ℓ p 1)) θ or (B(L p 0), B(L p 1)) θ for any pair 1 ≤ p 0 , p 1 ≤ ∞ and 0 < θ < 1. In the same vein, given a locally compact Abelian group G, let M (G) (resp. P M (G)) be the space of complex measures (resp. pseudo-measures) on G equipped with the usual norm µ M (G) = |µ|(G) (resp. µ P M (G) = sup{|ˆµ(γ)| γ ∈ G}). We describe similarly the interpolation space (M (G), P M (G)) θ. Various extensions and variants of this result will be given, e.g. to Schur multipliers on B(ℓ 2) and to operator spaces.