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

Abstract

In this work we study the geometry of the unit ball of the space of operators $\mathcal{L}(X,Y^{\ast })$, 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 $x_0^{\ast }\otimes y_0^{\ast }$ in the unit sphere $S_{\mathcal{L}(X,Y^{\ast })}$ is a weak${}^{\ast }$-strongly extreme point of the unit ball, then $x_0^{\ast }$ is weak${}^{\ast }$-strongly extreme point of unit ball of $X^{\ast }$ and $y_0^{\ast }$ is weak${}^{\ast }$-strongly extreme point of the unit ball of $Y^{\ast }$. We show that a similar conclusion holds if the rank one operator is a Namioka point (equivalently, point of weak${}^{\ast }$-weak continuity for the identity mapping) on the unit sphere of $\mathcal{L}(X,Y^{\ast })$. We also study extremal phenomenon in the unit ball of $\mathcal{L}(X,Y^{\ast }{)}^{\ast }$. 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 $z\in S_{\mathcal{L}(X,Y^{\ast }{)}^{\ast }}$ is a weak${}^{\ast }$-strongly extreme point of the unit ball, then $z=x\otimes y$ for some weak${}^{\ast }$-strongly extreme points $x\in S_X$ and $y\in S_Y$, provided the space of compact operators, $\mathcal{K}(X,Y^{\ast })$ is separating for $X{\hat{\otimes }}_{\pi }Y$.