On characterizing the standard quantum logics
W. John Wilbur · Transactions of the American Mathematical Society · 1977
Let L \mathcal {L} be a complete projective logic. Then L \mathcal {L} has a natural representation as the lattice of ⟨ ⋅ , ⋅ ⟩ \langle { \cdot , \cdot } \rangle -closed subspaces of a left vector space V over a division ring D , where ⟨ ⋅ , ⋅ ⟩ \langle {\cdot ,\cdot } \rangle is a definite θ \theta -bilinear symmetric form on V , θ \theta being some involutive antiautomorphism of D . Now a well-known theorem of Piron states that if D is isomorphic to the real field, the complex field or the sfield of quaternions, if θ \theta is continuous, and if the dimension of L \mathcal {L} is properly restricted, then L \mathcal {L} is just one of the standard Hilbert space logics. Here we also assume L \mathcal {L} is a complete projective logic. Then if every θ \theta -fixed element of D is in the center of D and can be written as ± d θ ( d