Positive representations of $C_0(X)$. 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 $\pi$ 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.Comment: There is now a direct proof of the existence of a generating regular positive spectral measure in the case of KB-spaces, without resorting to the Banach space theory. References to the existing literature on the Banach space case have been added, and perspectives for future research are now given. 24 pages, to appear in Ann. Funct. Ana