turkmath.org

Isometric bipositive representations of normed real ordered spaces with a closed cone on $C(X)$ with $X$ a compact Hausdorff space is a classical subject that has been considered by many authors and there are well-known characterizations of such representability in the literature. We consider the more general case of locally convex real ordered space with a specified family of seminorms, and through Schaefer's construction obtain characterizations of isometric bipositive representability of such spaces on $C(X)$, with $X$ a locally compact Hausdorff space. Some characterizations are new even in the normed case. In particular, we single out a relation of such spaces with locally $C^{*}$-algebras.