The operator Hilbert space \(OH\), complex interpolation and tensor norms.

*(English)*Zbl 0932.46046
Mem. Am. Math. Soc. 585, 103 p. (1996).

The theory of operator spaces studies various closed subspaces of the algebra \(B(H)\) of all bounded operators in a Hilbert space \(H\), and completely bounded maps between them. In vague terms, a completely bounded (respectively, completely isometric) map between two operator spaces is a bounded linear operator (respectively, a linear isometry) that “remembers much” about the way the spaces are embedded in \(B(H)\). We refer the reader to the papers of E. G. Effros and Z.-J. Ruan [Can. Math. Bull. 34, No. 3, 329-337 (1991; Zbl 0769.46037)] and of D. P. Blecher and V. J. Paulsen [J. Funct. Anal. 99, No. 2, 262-292 (1991; Zbl 0786.46056)] for the precise definitions and basic facts.

It happens quite often that some subspaces of \(B(H)\) are isomorphic (or even isometric) as Banach spaces, but are different as operator spaces (i.e., are not completely isomorphic). The Hilbert space \(\ell^2\) has a continuum of pairwise different operator space realizations; in particular, the most well-known “row” and “column” realizations \(R\) and \(C\) are different. Unlike the situation in Banach space theory, \(R\) and \(C\), as well as many other “classical” operator Hilbert spaces fail to be self-dual in the operator sense. In particular, \(R^* =\overline C\), and vice versa.

In the book under review it is shown that in each dimension (finite or infinite) there is an operator Hilbert space \(K\) equal to its antidual (“equal” means that the canonical Riesz identification of \(K^*\) and \(\overline K\) is a complete isometry) and that, up to complete isometry, this space is unique. The separable infinite dimensional space of this type is denoted by \(OH\). In a sense, \(OH\) occupies a “central” position among operator spaces (like the Hilbert space among Banach spaces). Along this guideline, the author proves, among other things, that the complex interpolation formula \(OH=({\overline E}^*, E)_{1/2}\) is true for any operator space \(E\) with some natural and mild additional structure (in particular, \(OH=(C, R)_{1/2}\)). By anology with Banach space theory, the linear maps that factor through \(OH\) are studied, along with the related tensor products. An operator space version is proved of the Fritz John theorem on the maximal volume ellipsoid.

The above list is incomplete, and the “analogy scheme” presented in this review is a bit deceiving. From the operator space theory viewpoint, distinction is more important, and intrinsic results, various ramifications of the subject, applications, intermediate technical notions, and specific details described in the book are the most interesting. Though extensive quotation of all that would require too much background, I want to emphasize that the book is in fact addressed to a much wider audience than merely the experts in operator spaces: it is sufficiently comfortable for nonexperts.

It happens quite often that some subspaces of \(B(H)\) are isomorphic (or even isometric) as Banach spaces, but are different as operator spaces (i.e., are not completely isomorphic). The Hilbert space \(\ell^2\) has a continuum of pairwise different operator space realizations; in particular, the most well-known “row” and “column” realizations \(R\) and \(C\) are different. Unlike the situation in Banach space theory, \(R\) and \(C\), as well as many other “classical” operator Hilbert spaces fail to be self-dual in the operator sense. In particular, \(R^* =\overline C\), and vice versa.

In the book under review it is shown that in each dimension (finite or infinite) there is an operator Hilbert space \(K\) equal to its antidual (“equal” means that the canonical Riesz identification of \(K^*\) and \(\overline K\) is a complete isometry) and that, up to complete isometry, this space is unique. The separable infinite dimensional space of this type is denoted by \(OH\). In a sense, \(OH\) occupies a “central” position among operator spaces (like the Hilbert space among Banach spaces). Along this guideline, the author proves, among other things, that the complex interpolation formula \(OH=({\overline E}^*, E)_{1/2}\) is true for any operator space \(E\) with some natural and mild additional structure (in particular, \(OH=(C, R)_{1/2}\)). By anology with Banach space theory, the linear maps that factor through \(OH\) are studied, along with the related tensor products. An operator space version is proved of the Fritz John theorem on the maximal volume ellipsoid.

The above list is incomplete, and the “analogy scheme” presented in this review is a bit deceiving. From the operator space theory viewpoint, distinction is more important, and intrinsic results, various ramifications of the subject, applications, intermediate technical notions, and specific details described in the book are the most interesting. Though extensive quotation of all that would require too much background, I want to emphasize that the book is in fact addressed to a much wider audience than merely the experts in operator spaces: it is sufficiently comfortable for nonexperts.

Reviewer: S.V.Kislyakov (St.Peterburg)

##### MSC:

46L07 | Operator spaces and completely bounded maps |

47L25 | Operator spaces (= matricially normed spaces) |

46B70 | Interpolation between normed linear spaces |

46M05 | Tensor products in functional analysis |

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |

46L06 | Tensor products of \(C^*\)-algebras |