On the de Morgan property of the standard Brouwer-Zadeh poset

The standard Brouwer-Zadeh poset Σ(H) is the poset of all effect operators on a Hilbert space H, naturally equipped with two types of orthocomplementation. In developing the theory, the question occured if (when) Σ(H ) fulfils the de Morgan property with respect to both orthocomplementation operations. In Ref. 3 the authors proved that it is the case provided dim H < ∞, and they conjectured that if dim H = ∞, then the answer is in the negative. In this note, we first give a somewhat simpler proof of the known result for dim H < ∞, and then we give a proof to the conjecture: We show that if dim H = ∞, then the de Morgan property is not valid