A characterization of the second dual of $C_ 0(S,A)$

Abstract:

Let $S$ be a locally compact Hausdorff space, and let $A$ be a Banach space. The space of the continuous functions from $S$ to $A$ vanishing at infinity is denoted by ${C_0}(S,A)$. Let $MW(S,{A^{\ast }})$ be the space of the representing measures of all the bounded linear functionals on ${C_0}(S,A)$. For $\mu \in MW(S,{A^{\ast }})$ let \[ {L_\infty }(|\mu |,{A^{{\ast }{\ast }}},{A^{\ast }}) = \{ f:S \to {A^{{\ast }{\ast }}}|f( \cdot ){x^{\ast }} \in {L_\infty }(|\mu |)\forall {x^{\ast }} \in {A^{\ast }}\}.\] The second dual of ${C_0}(S,A)$ is characterized in the general case by means of certain elements in the product linear space $\prod {\{ {L_\infty }(|\mu |,{A^{{\ast }{\ast }}},{A^{\ast }}):\mu \in MW(S,{A^{\ast }})\} }$.