Notation: Let $K$ be a compact Hausdorff space. Denote $C(K)$ to be the set of all scalar-valued continuous functions on $K.$
In the book 'Isometries on Banach Spaces: Vector-Valued Function Spaces and Operator Spaces' by Fleming and Jamison, page $18,$ they gave a definition of a function module.
Definition: A Banach function module or function module is a triple $(K,(Y_t)_{t\in K}, Y),$ where $K$ is a nonempty compact Hausdorff space (the base space), $(Y_t)_{t\in K}$ is a family of Banach spaces (the component spaces), and $Y$ a closed subspace of $\prod_{t\in K}Y_t$ such that the following conditions are satisfied:
(i) $hy\in Y$ for $y\in Y$ and $h\in C(K)$
(ii) $t\mapsto \|y(t)\|$ is an upper semicontinuous function for every $y\in Y,$
(iii) $Y_t=\{y(t):y\in Y\},$
(iv) $\overline{\{t\in K:Y_t\neq \{0\}\}} = K$
Let $E$ be a Banach space and $L$ be a locally compact space. Denote $C_b(L,E)$ to be the set of all $E$-valued bounded continuous functions on $L.$
Question: Is $C_b(L,E)$ a function module?
Based on Fleming and Jamison, if $L$ is not compact, one can take its Stone-Cech compactification $\beta L$ and $Y_t = E$ if $t\in L$ and $Y_t=\{0\}$ if $t\in \beta L \setminus L.$
However, I do not know that the compactified space is a function module or not.