Weakly Compact Operators Whose Adjoints Preserve $C^*$-Structure

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