[Author]
 Oleg Reinov

[Title]
 Grothendieck's inequality and applications

[AMS Subj-class]
 46B28 Spaces of operators; tensor products; approximation properties

[Abstract]
 The famous Grothendieck inequality, which can be seen as
 a matrix inequality associated to certain bilinear operators and
 called by him "the fundamental theorem
 of the metric theory of tensor products of Banach spaces",
 is equivalent to the following assertion:

 {\it
 Let $\{a_{ij}\}_{i,j=1}^n$ be a finite matrix of real numbers
 such that $|\sum_{i,j=1}^n a_{ij}t_is_j|\le 1$
 whenever $|t_i|, |s_j|\le1.$ Then for every set of unit vectors
 $\{x_i\}_{i=1}^n$ and $\{y_j\}_{j=1}^n$ in a Hilbert space
 $\|\sum_{i,j} a_{ij}(x_i,y_j)\|\le K,$ where $K$ is an absolute constant.
 }\

 This theorem has a lot of generalizations and applications in very
 different directions. Some of them are investigations of multilinear
 extensions of the inequality as well as considerations of the cases
 of so-called operator spaces and of non-commutative $L_p$-spaces
 (such as the Schatten spaces $S_p).$
 Let us mention just a few fields of applications:

 $\bullet$\ Theory of absolutely $p$-summing operators with application to
 the isomorphic classification of Banach spaces and to the geometry of normed
 spaces in general (an example: disk algebra $C_A(\Bbb T)$ is not
 isomorphic to a factor space of a $C(K)$-space);

 $\bullet$\ Investigations of  uniform Banach algebras and, generally, of
 $Q$-algebras (commutative Banach algebras which are isomorphic as Banach
 algebras to the quotients of uniform Banach algebras). An example:
 the answers (for $1\le p\le \infty)$ to the old (essentially, due to Varopoulos)
 problem whether $S_p$-spaces (with their Schur products) should be $Q$-algebras;

 $\bullet$\ Problems of vector measures theory and related questions in
 geometric theory of Banach spaces (such as constructions of
 counterexamples to some long standing problems. For instance,
 to the question of whether a separable Banach space
 does not contain $l_1$ if and only if its dual space is separable).

 We shall be concerned with just some small (but hope, ones of the main) parts
 of the topic in connection with this beautiful Grothendieck inequality.

[Comments]
 LaTeX, English, 10 pp.

[Contact e-mail]
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