Foundations of Physics

, Volume 30, Issue 10, pp 1801–1805 | Cite as

On the de Morgan Property of the Standard Brouwer–Zadeh Poset

  • G. Cattaneo
  • J. Hamhalter
  • P. Pták


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.


Hilbert Space Simple Proof Effect Operator 
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Copyright information

© Plenum Publishing Corporation 2000

Authors and Affiliations

  • G. Cattaneo
    • 1
  • J. Hamhalter
    • 2
  • P. Pták
    • 3
  1. 1.Dipartimento di Informatica, Sistemistica e ComunicazioneUniversità di Milano-BicoccaMilanoItaly;
  2. 2.Department of Mathematics, Faculty of Electrical EngineeringCzech Technical UniversityPraha 6Czech Republic;
  3. 3.Department of Mathematics, Faculty of Electrical EngineeringCzech Technical UniversityPraha 6Czech Republic;

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