Representation of strongly truncated Riesz spaces

TL;DR

This paper introduces strongly truncated Riesz spaces with a spectrum, showing they can be represented as dense subspaces of continuous functions, generalizing classical representation theorems in Riesz space theory.

Contribution

It defines strongly truncated Riesz spaces and proves a new representation theorem generalizing Kakutani's classical result, linking these spaces to $C_0(X)$-spaces.

Findings

Strongly truncated Riesz spaces are isomorphic to dense subspaces of $C_0(X)$-spaces.

The main theorem generalizes Kakutani's representation theorem.

A new proof of Fremlin's representation theorem is derived.

Abstract

Following a recent idea by Ball, we introduce the notion of strongly truncated Riesz space with a suitable spectrum. We prove that, under an extra Archimedean type condition, any strongly truncated Riesz space is isomorphic to a uniformly dense Riesz subspace of a $C_{0}\left( X\right) $-space. This turns out to be a direct generalization of the classical Kakutani Representation Theorem on Archimedean Riesz spaces with strong unit. Another representation theorem on normed Riesz spaces, due to Fremlin, will be obtained as a consequence of our main result.