Positive representations of $C_0(X)$. I

TL;DR

This paper introduces positive spectral measures on sigma-algebras that generate positive representations of $C_0(X)$ on Banach lattices, establishing a unique correspondence in the context of KB-spaces and exploring order-specific phenomena.

Contribution

It defines positive spectral measures and demonstrates their role in representing $C_0(X)$ on Banach lattices, including uniqueness results for KB-spaces.

Findings

Positive spectral measures generate positive representations of $C_0(X)$.

On KB-spaces, such representations are uniquely generated by a regular positive spectral measure.

Order-specific phenomena influence the relationship between representations and spectral measures.

Abstract

We introduce the notion of a positive spectral measure on a $\sigma$-algebra, taking values in the positive projections on a Banach lattice. Such a measure generates a bounded positive representation of the bounded measurable functions. If $X$ is a locally compact Hausdorff space, and $\pi$ is a positive representation of $C_0(X)$ on a KB-space, then $\pi$ is the restriction to $C_0(X)$ of such a representation generated by a unique regular positive spectral measure on the Borel $\sigma$-algebra of $X$. The relation between a positive representation of $C_0(X)$ on a Banach lattice and -- if it exists -- a generating positive spectral measure on the Borel $\sigma$-algebra is further investigated; here and elsewhere phenomena occur that are specific for the ordered context.