Weakly Compact Operators Whose Adjoints Preserve $C^{\ast }$-Structure

Abstract

Let $\Omega$ be a compact Hausdorff space and $X$ be a complex Banach space such that $X^{\ast \ast }$ is isometric to a $C^{\ast }$-algebra. Let $\mathcal{W}(X^{\ast },C(\Omega ))$ denote the space of weakly compact operators. In this article, we show that the collection of operators $T$ in $\mathcal{W}(X^{\ast },C(\Omega ))$ such that $T^{\ast }$ maps linear extreme points of the unit ball of $C(\Omega {)}^{\ast }$ to $C^{\ast }$-extreme points of $X^{\ast \ast }$ is a uniformly strongly extreme set. We also investigate this phenomenon when $X^{\ast \ast }$ is isometric to a space of all operators on a Banach space.

11 pages. Comments are welcome