I've been using the representation result below, from Krasnosel'skij/Lifshits/Sobolev; Positive Linear Systems---The Method of Positive Linear Operators. Heldermann Verlag, 1989.
Theorem. Let $E$ be a real Banach space partially ordered by a solid, normal and minihedral cone $E_+$. Then there exist a compact Hausdorff space $Q$ and a linear homeomorphism $\Phi: E \rightarrow C(Q)$ such that $\Phi(E_+) = C_+(Q)$.
In this statement, $C(Q)$ stands for the space of continuous, real-valued functions on $Q$, while $C_+(Q)$ consists of the nonnegative ones. This is stated as Theorem 6.6 on page 64 in the book.
I'm wondering if anybody knows of a better presentation of this result. It's not the first time that I've been having problems with this book, which has many mistakes. The original paper of Kakutani from 1941 doesn't seem much easier to read, and Google searches haven't yielded much on that direction.