Extremal Structure of Projective Tensor Products

Abstract

We prove that, given two Banach spaces X and Y and bounded, closed convex sets C ⊆ X and D ⊆ Y , if a nonzero element z ∈ co(C ⊗ D) ⊆ X ⊗πY is a preserved extreme point then z = x0 ⊗ y0 for some preserved extreme points x0 ∈ C and y0 ∈ D, whenever K(X, Y ∗ ) separates points of X ⊗πY (in particular, whenever X or Y has the compact approximation property). Moreover, we prove that if x0 ∈ C and y0 ∈ D are weak-strongly exposed points then x0 ⊗ y0 is weak-strongly exposed in co(C ⊗D) whenever x0 ⊗y0 has a neighbourhood system for the weak topology defined by compact operators. Furthermore, we find a Banach space X isomorphic to 2 with a weak-strongly exposed point x0 ∈ BX such that x0 ⊗x0 is not a weak-strongly exposed point of the unit ball of X ⊗πX.