Spectral resolution in an order-unit space

The operational approach to quantum physics employs an order-unit space in duality with a base-normed space, and in this context, a suitable spectral theory is a prerequisite for the representation of quantum-mechanical observables. An order-unit space is called spectral if it is enriched by a compression base with the comparability and projection cover properties. These notions are explicated in the article. We show that each element in a spectral order-unit space determines and is determined by a spectral resolution and it has a spectrum which is a nonempty closed bounded subset of the real numbers. Our theory is a generalization and a more algebraic version of the well-known non-commutative spectral theory of Alfsen and Shultz.