New advances on the Grothendieck's inequality problem for bilinear forms on JB*-triples

Abstract

The results known as Grothendieck’s inequalities began with the famous paper [8] in which A. Grothendieck proved the so-called “Grothendieck’s inequalities” for commutative C*-algebras. These inequalities were generalized by G. Pisier [18] and U. Haagerup [10, 9] to the setting of C*-algebras. Every C*-algebra belongs to a more general class of Banach spaces known as JB*-triples (see definition and examples below). JB*-triples were introduced by Kaup [14] in the study of bounded symmetric domains in complex Banach spaces. The class of JB*-triples has been intensively developed in the last twenty years. In the setting of JB*-triples, Grothendieck’s inequalities were studied by T. Barton and Y. Friedman [1], C.-H. Chu, B. Iochum and G. Loupias [3], A. M. Peralta [15] and A. M. Peralta and A. Rodŕiguez Palacios [16, 17]. The natural prehilbertian seminorms associated derived from states in a C*-algebra do not make sense in a JB*-triple because the latter needs not have, in general, a natural order structure. In the setting of JB*-triples, the prehilbertian seminorms associated to norm-one functionals are constructed as follows: Let φ be a norm-one element in the dual space of a JB*-triple