Representation of Banach spaces partially ordered by solid, normal, minihedral cones

Question

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.

Thanks!

Answer 1

I think Kakutani's theorem is presented in Lindenstrauss and Tzafriri's "Classical Banach spaces, II" and Lacey's "The isometric theory of the classical Banach spaces", neither of which I have at hand right now. It is included in Peter Meyer-Nieberg's "Banach lattices". Kakutani's theorem is beautiful and useful but not very difficult given standard material in a real analysis course.

Answer 2

Most of the proofs of this result are stated in terms of Banach lattices rather than in terms of solid, normal and minihedral cones; this might explain why Google searches are being unfriendly.

In any case, here are two wonderful references (different from the ones indicated by Bill Johnson):

H. H. Schaefer - Banach Lattices and Positive Operators, Springer 1975

All you need here is Chapter 2 called Banach Algebras (and there's no need to read Chapter 1 for your purposes). Schaefer's book defines the positive cone terminology initially in the chapter, but instead of developing properties of the cone (minihedral, etc) it goes straight to the properties of the resulting lattice.

A more modern and infinitely less cone-y version can be found in my new favorite book for such matters:

Aliprantis and Burkinshaw - Positive Operators, Springer (2006)

Your result is Theorem 4.29, and sadly it requires much material from the first three chapters.