Extreme operators in the unit ball of $L(C(X),C(Y))$ over the complex field

Abstract:

Assume that $X$ and $Y$ are compact Hausdorff spaces and that $C(X)$ and $C(Y)$ are the Banach spaces of continuous complex-valued functions on $X$ and $Y$, respectively. $L(C(X),C(Y))$ is the space of bounded linear operators from $C(X)$ to $C(Y)$. If $E$ is a Banach space, then $S(E)$ is the closed unit ball in $E$. An operator $T$ in $S(L(C(X),C(Y)))$ is nice if ${T^ \ast }(\operatorname {ext} S(C{(Y)^ \ast })) \subset \operatorname {ext} S(C{(X)^ \ast })$. For each $y \in Y,{\varepsilon _y}$ denotes point mass at $y$. The main theorem states that if $T$ is extreme in $S(L(C(X),C(Y)))$ and $||{T^ \ast }({\varepsilon _y})|| = 1$ for all $y \in Y$, then $T$ is nice. Other theorems are proved by using the same techniques as in the proof of the main theorem.