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The metric theory of tensor products. Grothendieck’s résumé revisited. (English) Zbl 1186.46004
Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4440-3/hbk). x, 278 p. (2008).
Like most functional analysts of my generation, I had learnt basic tensor product theory and its applications to vector measures from the now classic monograph of J. Diestel and J. J. Uhl [“Vector measures” (AMS Mathematical Surveys 15) (1977; Zbl 0369.46039)]. More than three decades hence, this latest offering by Joe Diestel, Jan H. Fourie and Johan Swart, gives, as always, a very readable, up to date and comprehensive account of the tensor product theory, as originated from the celebrated ‘Résumé’ of Grothendieck – a highly commendable effort indeed. To quote from the Preface, “The ‘Résumé’ sets forth Grothendieck’s plan for the study of the finer structure of Banach spaces. He uses tensor products as a foundation upon which he builds the classes of operators most important to the study and establishes the importance of the ‘local theory’ in the study of these operators and the spaces they act upon.”
I have not read the ‘Résumé’, but had, over the years, listened to and read expositions of several parts of it by experts. My first exposure, as a fresh PhD, was, when I attended Gilles Pisier’s mini-course at a conference in Missouri-Columbia in 1984. This book nicely fills the gap in one’s education by putting things in perspective.
Chapter 1 recalls the basics of reasonable cross norms $$\alpha$$ (tensor norms) with particular emphasis on the injective and projective norms. The fundamental factorization theorem for integral operators and “the remarkable role” of $$C(K)$$ and $$L^1$$ spaces gets demonstrated in Chapter 2. In particular, after introducing the left/right, injective/projective hulls of $$\alpha$$, Grothendieck’s theorem that ‘such norms’ coincide with $$\alpha$$ if one of the component spaces is a $$C(K)$$ space or $$L^1$$ space is proved. The chapter ends with a table of natural tensor norms and a description of the factorization scheme that characterizes the associated integral operators and their position in the lattice ordering of tensor norms.
Taking on Hilbertian notions, the tensor norms $$H$$ (symmetric and injective) and $$H^\ast$$ (symmetric and projective) are investigated in the early part of Chapter 3. A highlight of this chapter is the ‘little’ Grothendieck inequality: there is a constant $$\sigma$$ such that for any Hilbert space $$H$$, $$\|\text{Id}_H\|_{/H^\ast \backslash} \leq \sigma$$. Equality holds if and only if $$H$$ is infinite-dimensional. In the case of a real Hilbert space, $$\sigma = \frac{\pi}{2}$$, and $$\sigma = \frac {4}{\pi}$$ in the complex case.
The crowning glory is Chapter 4 where the Fundamental Theorem (Grothendieck’s inequality) and its consequences are considered: There is a universal constant $$K_G$$ such that for any Hilbert space $$H$$, $$\|\text{Id}_H\|_{/\wedge \backslash} \leq K_G$$, where for real scalars, $$K_G \leq \text{sinh}(\frac{\pi}{2})$$ and, for the complex case, $$K_G \leq 2\text{sinh}(\frac{\pi}{2})$$. The least constant for which this inequality holds is called Grothendieck’s constant. Its precise value in either scalar field is not known. The proof for the best known bound for $$K_G(\mathbb R)$$, due to Krivine, gets into Appendix A.
A theorem of Carne that gives, for Banach algebras $$A,B$$, equivalent conditions (not involving Banach algebras!) under which the natural algebra structure on $$A \otimes B$$ induces a Banach algebra structure on $$A \otimes^{\alpha} B$$, is also in this chapter.
The fourteen tensor norms, considered thus far, are not equivalent. And according to a result of Pérez-García and Villanueva, these norms do not preserve lattice structure under tensor products.
The problems from the ‘Résumé’, with an abridged account of their solution or current status, gets discussed in Appendix A. Appendix B has the Blaschke selection principle and compact convex sets in finite-dimensional Banach spaces. A short introduction to Banach lattices, leading to the proof due to Krivine and Carne of Grothendieck’s inequality in the Banach lattice setting, is found in Appendix C. Appendix D introduces injective Banach spaces and an Epilogue discusses these matters for operator spaces.
A glossary of terms, index of notation, in addition to a subject and an author index, make the monograph easy to use. There is an extensive bibliography, not numbered, but papers get referred to by authors’ names and year of publication.
This review was written during a visit to the Chennai Mathematical Institute, whom the reviewer thanks for hospitality.

##### MSC:
 46-02 Research exposition (monographs, survey articles) pertaining to functional analysis 46B03 Isomorphic theory (including renorming) of Banach spaces 46B04 Isometric theory of Banach spaces 46B28 Spaces of operators; tensor products; approximation properties 47L20 Operator ideals