Positive representations of C0(X), I

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 if $\pi$ is a positive representation of $C_0\left(X\right)$ on a KB-space, then $\pi$ is the restriction to $C_0\left(X\right)$ 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 $C_0\left(X\right)$ on a Banach lattice and—if it exists—a generating positive spectral measure on the Borel $\sigma$-algebra are further investigated; here and elsewhere, phenomena occur that are specific for the ordered context.