The Grothendieck inequality revisited (Q2925657)

Grothendieck's inequality is a profound contribution of \textit{A. Grothendieck} to Banach space theory and operator theory, published in his famous [``Résumé de la théorie métrique des produits tensoriels topologiques'', Bol. Soc. Mat. Sao Paulo 8, 1--79 (1956; Zbl 0074.32303)]. This result is known as `the fundamental theorem in the metric theory of tensor products'. In 1968, \textit{J. Lindenstrauss} and \textit{A. Pełczyński} [Stud. Math. 29, 275--326 (1968; Zbl 0183.40501)] reformulated the statement of Grothendieck's inequality to the following: There exists a positive constant \(K_{G}\) such that, for all Hilbert spaces \(H\), for all positive integers \(n\), every scalar matrix \(\left( a_{ij}\right) _{n\times n}\) and all \( x_{1},\dots,x_{n},\) \(y_{1},\dots,y_{n}\) in the unit ball of \(H\), we have NEWLINE\[NEWLINE \left| \sum_{i,j}a_{ij}\langle x_{i},y_{j}\rangle \right| \leq K_{G}\sup \left\{ \left| \sum_{i,j}a_{ij}s_{i}t_{j}\right| :\left| s_{i}\right| ,\left| t_{j}\right| \leq 1\right\} . NEWLINE\]NEWLINE This inequality has many different applications in several different directions, and still has challenging related open questions. For instance, a striking advance in the investigation of the constant \(K_{G}\) (Grothendieck's constant) was recently obtained by \textit{M. Braverman} et al. [Forum Math. Pi 1, Article ID e4 (2013; Zbl 1320.15016)], disproving an old conjecture of Krivine on the exact value of \(K_{G}\).NEWLINENEWLINEIn this monograph, the author considers the classical Grothendieck inequality as a statement about representations of functions of two variables over discrete domains by integrals of two-fold products of functions of one variable. Some variants of the Grothendieck inequality in higher dimensions are derived and multilinear Parseval-like formulas are obtained, resulting in multilinear extensions of the Grothendieck inequality, and some other consequences. For instance, the results are used to characterize the feasibility of integral representations of multilinear functionals on a Hilbert space.