Isometries of spaces of weak $*$ continuous functions

Abstract:

If $X$ is a compact Hausdorff space and ${E^*}$ a Banach dual, we denote by $C(X,({E^*},{\sigma ^*}))$ the Banach space of continuous functions from $X$ to ${E^*}$ when the latter space is given its weak * topology. It is shown that if $E_1^*,E_2^*$ have trivial centralizers and satisfy a topological condition introduced by Namioka and Phelps, then, given an isometry $T$ mapping $C(X_1^*,({E_1},{\sigma ^*}))$ onto $C({X_2},(E_2^*,{\sigma ^*}))$ and given $F \in C({X_1},(E_1^*,{\sigma ^*}))$, there exists a dense ${G_\delta }$ in ${X_2}$ on which \[ (*)\quad (TF)(x) = U(x)F \circ \Phi (x),\] where $\Phi$ is a homeomorphism of ${X_2}$ onto ${X_1}$ and $U( \cdot )$ an operator-valued function independent of $F$. If the $E_i^*$ are separable then the $U(x)$ are surjective isometries and $x \to U(x)$ is continuous. When the ${X_i}$ are metric the representation (*) holds on all of ${X_2}$.