From effect algebras to algebras of effects (sum Brouwer Zadeh algebras)

The algebraic structures arising in the axiomatic framework of unsharp quantum mechanics based on effect operators on a Hilbert space are investigated. It is stressed that usually considered "effect" algebras neglect the unitary Brouwerian map of complementation, and the main results based on this complementation are collected, showing the enrichment produced into the theory by its introduction. In particular, in these structures two notions of sharpness can be considered: K-sharpness induced by the usual complementation of effect algebras and B-sharpness induced by this new complementation. Quantum (resp., classical) SBZ algebras are then characterized by the condition of B-coherence (resp., B-coherence plus B-compatibility), showing that in this case the poset of all B-sharp elements is orthomodular (resp., Boolean algebra). In the unsharp context of effect operators, the finite dimensionality of the Hilbert space or the finiteness of a von Neumann algebra are both characterized by a de Morgan property of the Brouwer complementation. Moreover, since effect operators on a pre-Hilbert space give rise to a standard model of effect algebras, a characterization of completeness of pre-Hilbert spaces is given making use of the Brouwer complement