Extremal structure of projective tensor products

TL;DR

This paper characterizes the extremal structure of projective tensor products of Banach spaces, linking preserved extreme points to tensor products of such points and exploring weak-strong exposure properties.

Contribution

It establishes conditions under which preserved extreme points of tensor products are tensor products of preserved extreme points, and analyzes weak-strong exposure in tensor product spaces.

Findings

Preserved extreme points in tensor products are tensor products of preserved extreme points under certain conditions.

Weak-strongly exposed points in tensor products retain their property under specific topological conditions.

An example shows that tensor products of weak-strongly exposed points may fail to be weak-strongly exposed.

Abstract

We prove that, given two Banach spaces $X$ and $Y$ and bounded, closed convex sets $C\subseteq X$ and $D\subseteq Y$, if a nonzero element $z\in \overline{\mathrm{co}}(C\otimes D)\subseteq X\widehat{\otimes}_\pi Y$ is a preserved extreme point then $z=x_0\otimes y_0$ for some preserved extreme points $x_0\in C$ and $y_0\in D$, 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 $x_0\in C$ and $y_0\in D$ are weak-strongly exposed points then $x_0\otimes y_0$ is weak-strongly exposed in $\overline{\mathrm{co}}(C\otimes D)$ whenever $x_0\otimes y_0$ 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 $x_0\in B_X$ such that $x_0\otimes x_0$ is not a weak-strongly exposed point of the unit ball of $X\widehat{\otimes}_\pi X$.