The expectation monad in quantum foundations

The expectation monad is introduced and related to known monads: it sits between on the one hand the distribution and ultrafilter monad, and on the other hand the continuation monad. The Eilenberg-Moore algebras of the expectation monad are characterized as convex compact Hausdorff spaces, using a theorem of Świrszcz. These convex compact Hausdorff spaces are dually equivalent to Banach (complete) order unit spaces, via a result of Kadison, which in turn are equivalent to Banach effect modules. In this way we obtain a close 'triangle' relationship between predicates and states for the expectation monad. Moreover, the approach leads to a new reformulation of Gleason's theorem, expressing that effects on a Hilbert space are free effect modules on projections, obtained via tensoring with the unit interval.