Metrical aspects of the complexification of tensor products and tensor norms

Abstract

We study the relationship between real and complex tensor norms. The theory of tensor norms on tensor products of Banach spaces, was developed, by A. Grothendieck, in his Resumé de la théorie métrique des produits tensoriels topologiques [3]. In this monograph he introduced a variety of ways to assign norms to tensor products of Banach spaces. As is usual in functional analysis, the real-scalar theory is very closely related to the complex-scalar theory. For example, there are, up to top ological equivalence, fourteen ``natural' tensor norms in each of the real-scalar and complex-scalar theories. This correspondence was remarked upon in the Resumé, but without proving any formal relationships, although hinting at a certain injective relationship between real and complex (topological) equivalence classes of tensor norms. We make explicit connections between real and complex tensor norms in two different ways. This divides the dissertation into two parts. In the first part, we consider the ``complexifications' of real Banach spaces and find tensor norms and complexification procedures, so that the complexification of the tensor product, which is itself a Banach space, is isometrically isomorphic to the tensor product of the complexifications. We have results for the injective tensor norm as well as the projective tensor norm. In the second part we look for isomorphic results rather than isometric. We show that one can define the complexification of real tensor norm in a natural way. The main result is that the complexification of real topological equivalence classes that is induced by this definition, leads to an injective correspondence between the real and the complex tensor norm equivalence classes.