On the de Morgan Property of the Standard Brouwer–Zadeh Poset
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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 dimH<∞, and they conjectured that if dimH=∞, then the answer is in the negative. In this note, we first give a somewhat simpler proof of the known result for dimH<∞, and then we give a proof to the conjecture: We show that if dimH=∞, then the de Morgan property is not valid.
KeywordsHilbert Space Simple Proof Effect Operator
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