Quantum Logics and Convex Spaces

An orthomodular σ-lattice with rich set ofstates satisfying the property that every affinefunctional from the set of states into the unit intervalof the reals corresponds to an expectational functional of exactly one real observable (so-calledu-spectral logic) is compared with the noncommutativespectral theory of Alfsen and Shultz. Necessary andsufficient conditions are found under which these two approaches are in correspondence.