Geometry of the unit ball of ${\mathcal L}(X,Y^*)$

TL;DR

This paper investigates the geometric structure of the unit ball in operator spaces, focusing on extreme points and Namioka points, and establishes conditions under which elementary tensors inherit extremal properties.

Contribution

It characterizes when elementary tensors are extreme or Namioka points in the unit ball of ${ m L}(X,Y^*)$ and its dual, advancing understanding of the geometry of operator spaces.

Findings

Elementary tensors that are weak*-strongly extreme points have components that are also weak*-strongly extreme.

Similar results hold for Namioka points, linking extremal properties of tensors to their factors.

Under certain conditions, extremal points in the dual space are elementary tensors of extremal points in the original spaces.

Abstract

In this work we study the geometry of the unit ball of the space of operators ${\mathcal L}(X,Y^*)$, by considering the projective tensor product $X\hat{\otimes}_{\pi} Y$ as a predual. We prove that if an elementary tensor (rank one operator) of the form $x_0^*\otimes y_0^* $ in the unit sphere $ S_{{\mathcal L}(X,Y^*)}$ is a weak$^*$-strongly extreme point of the unit ball, then $x_0^*$ is weak$^*$-strongly extreme point of unit ball of $X^*$ and $y_0^*$ is weak$^*$-strongly extreme point of the unit ball of $Y^*$. We show that a similar conclusion holds if the rank one operator is a Namioka point (equivalently, point of weak$^*$-weak continuity for the identity mapping) on the unit sphere of ${\mathcal L}(X,Y^*)$. We also study extremal phenomenon in the unit ball of ${\mathcal L}(X,Y^*)^*$. We partly solve the open problem, when does an elementary tensor, whose components are Namioka points is again a Namioka point? We show that if a point $z\in S_{{\mathcal L}(X,Y^*)^*}$ is a weak$^*$-strongly extreme point of the unit ball, then $z=x\otimes y$ for some weak$^*$-strongly extreme points $x\in S_X$ and $y\in S_Y$, provided the space of compact operators, $\mathcal{K}(X,Y^*)$ is separating for $X\hat{\otimes}_{\pi} Y$.