Extremal Structure of Projective Tensor Products

Abstract

We prove that, given two Banach spaces X and Y and bounded, closed convex sets Csubseteq X and Dsubseteq Y, if a nonzero element zin {overline{{{,textrm{co},}}}}(Cotimes D)subseteq Xwidehat{otimes }_pi Y is a preserved extreme point then z=x_0otimes y_0 for some preserved extreme points x_0in C and y_0in D, whenever K(X,Y^*) separates points of X widehat{otimes }_pi Y (in particular, whenever X or Y has the compact approximation property). Moreover, we prove that if x_0in C and y_0in D are weak-strongly exposed points then x_0otimes y_0 is weak-strongly exposed in {overline{{{,textrm{co},}}}}(Cotimes D) whenever x_0otimes y_0 has a neighbourhood system for the weak topology defined by compact operators. Furthermore, we find a Banach space X isomorphic to ell _2 with a weak-strongly exposed point x_0in B_X such that x_0otimes x_0 is not a weak-strongly exposed point of the unit ball of Xwidehat{otimes }_pi X.