On characterizing the standard quantum logics

ABsTRAcr. Let e be a complete projective logic. Then e has a natural representation as the lattice of -closed subspaces of a left vector space V over a division ring D, where is a definite 0-bilinear symmetric form on V, 0 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 0 is continuous, and if the dimension of e is properly restricted, then e is just one of the standard Hilbert space logics. Here we also assume e is a complete projective logic. Then if every 0-fixed element of D is in the center of D and can be written as ? dG(d), some d E D, and if the dimension of e is properly restricted, we show that e is just one of the standard Hilbert space logics over the reals, the complexes, or the quaternions. One consequence is the extension of Piron's theorem to discontinuous 6. Another is a purely lattice theoretic characterization of the lattice of closed subspaces of separable complex Hilbert space.