The operator Hilbert space OH, complex interpolation and tensor norms
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Let H be any Hilbert space. We show that there is a Hilbert space H and an isometric embedding H B(H) into the space of all bounded operators on H, such that the canonical identification between H and its antidual (in the sense of the duality theory of operator spaces introduced by Blecher-Paulsen and Effros-Ruan) is a complete isometry. Moreover, the resulting operator space (which we call the operator Hilbert space) is unique up to complete isometry. When H = 2 we denote it by OH. We develop the theory of the operator space OH in connection with complex interpolation theory. We study the linear maps which factor through OH and the resulting tensor products in analogy with Banach space theory. Several reults there are valid not only for OH but for the more general class of Hilbertian homogeneous operator spaces which play an important rle throughout operator space theory. Finally, we initiate the "local" (i.e. finite dimensional) theory of operator spaces: we prove an operator space version of Fritz John's theorem on the maximal volume ellipsoid and several related estimates.