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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 $\vp\to Δ_X(\vp)$ tending to zero with $\vp>0$ such that every operator $T\colon L_2\to L_2$ with $\|T\|\le \vp$ that is simultaneously contractive (i.e. of norm $\le 1$) on $L_1$ and on $L_\infty$ must be of norm $\le Δ_X(\vp)$ on $L_2(X)$. We show that $Δ_X(\vp)\in O(\vp^α)$ 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 \ref{comcor4.3}), where $θ$-Hilbertian is meant in a slightly more general sense than in our previous paper \cite{P1}. 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 $T\colon L_2(μ)\to L_2(μ)$ belongs to this space iff $T\otimes id_X$ is bounded on $L_2(X)$ for any $θ$-Hilbertian space $X$. More generally, we are able to describe the spaces $$ (B(\ell_{p_0}), B(\ell_{p_1}))^θ{\rm or} (B(L_{p_0}), B(L_{p_1}))^θ$$ for any pair $1\le p_0,p_1\le \infty$ 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 $\|μ\|_{M(G)} = |μ|(G)$ (resp. \[ \|μ\|_{PM(G)} = \sup\{|\hatμ(γ)| \big| γ\in\hat G\}). \] 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 $B(\ell_2)$ and to operator spaces.