Quantum logics and convex spaces

An orthomodular sigma-lattice with rich set of states satisfying the property that every affine functional from the set of states into the unit interval of the reals corresponds to an expectational functional of exactly one real observable (so-called u-spectral logic) is compared with the noncommutative spectral theory of Alfsen and Shultz. Necessary and sufficient conditions are found under which these two approaches are in correspondence.