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 $\vp\to \Delta_X(\vp)$ 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 $\le \Delta_X(\vp)$ on $L_2(X)$. We show that $\Delta_X(\vp)\in O(\vp^\alpha)$ for some $\alpha>0$ iff $X$ is isomorphic to a quotient of a subspace of an ultraproduct of $\theta$-Hilbertian spaces for some $\theta>0$ (see Corollary \ref{comcor4.3}), where $\theta$-Hilbertian is meant in a slightly more general sense than in our previous paper \cite{P1}. Let $B_{r}(L_2(\mu))$ be the space of all regular operators on $L_2(\mu)$. We are able to describe the complex interpolation space $$(B_{r}(L_2(\mu), B(L_2(\mu))^\theta.$$ We show that $T\colon L_2(\mu)\to L_2(\mu)$ belongs to this space iff $T\otimes id_X$ is bounded on $L_2(X)$ for any $\theta$-Hilbertian space $X$. More generally, we are able to describe the spaces $$(B(\ell_{p_0}), B(\ell_{p_1}))^\theta {\rm or} (B(L_{p_0}), B(L_{p_1}))^\theta$$ for any pair $1\le p_0,p_1\le \infty$ and $0<\theta<1$. 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 $\|\mu\|_{M(G)} = |\mu|(G)$ (resp. $$\|\mu\|_{PM(G)} = \sup\{|\hat\mu(\gamma)| \big| \gamma\in\hat G\}).$$ We describe similarly the interpolation space $(M(G), PM(G))^\theta$. Various extensions and variants of this result will be given, e.g. to Schur multipliers on $B(\ell_2)$ and to operator spaces.