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

TL;DR

The paper analyzes the geometry of the unit ball of ${\mathcal L}(X,Y^*)$ through its predual $X\hat{\otimes}_{\pi} Y$, focusing on weak$^*$-strongly extreme points and Namioka points, and their behavior under projective tensor products. It proves stability results: a rank-one Namioka point $x_0^*\otimes y_0^*$ forces the components $x_0^*,y_0^*$ to be Namioka points, with partial converses under additional hypotheses, and extends these stability phenomena to higher duals. The work shows that a weak$^*$-strongly extreme point in higher duals must be an elementary tensor of weak$^*$-strongly extreme factors when a separating condition on ${\mathcal K}(X,Y^*)$ holds, with corollaries for finite-dimensional targets and unitary elements. It also addresses stability and density of points of continuity (weak-norm PC and Namioka points) under $M$-ideals and infinite codimension, yielding nonexistence results for many compact operators and clarifying nonhereditary aspects of Namioka density. Overall, the results deepen understanding of how tensor-product geometry controls extremal and continuity properties across duals of operator 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 $X\hat{\otimes}_π Y$ 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 $X\hat{\otimes}_π Y$.

Theorems & Definitions (30)

• Definition 1.1
• Theorem 1.2
• Theorem 2.1
• proof
• Corollary 2.2
• proof
• Lemma 2.3
• Theorem 2.4
• proof
• Proposition 2.5