This is an announcement for the paper "Complex interpolation between Hilbert, Banach and operator spaces" by Gilles Pisier.

Abstract: 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\widehat 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.

Archive classification: math.FA math.OA

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Submitted from: pisier@math.jussieu.fr

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