Complex Interpolation between Hilbert, Banach and Operator spaces

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Motivated by a question of Vincent Lafforgue, we study the Banach spaces X satisfying the following property: there is a function → Δₓ() tending to zero with >0 such that every operator T L₂→ L₂ with T≤ that is simultaneously contractive (i.e. of norm ≤ 1) on L₁ and on L∞ must be of norm ≤ Δₓ() on L₂(X). We show that Δₓ()∈ O(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 comcor4.3), where θ-Hilbertian is meant in a slightly more general sense than in our previous paper P1. 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 G equipped with the usual norm μₘ₍G₎ = |μ|(G) (resp. μPM(G) = {|μ(γ)| | γ∈ G}). We describe similarly the interpolation space (M(G), PM(G))θ. Various extensions and variants of this result will be given, e.g. to Schur multipliers on B(ℓ₂) and to operator spaces.