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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On characterizing the standard quantum logics


Author: W. John Wilbur
Journal: Trans. Amer. Math. Soc. 233 (1977), 265-282
MSC: Primary 81.06
DOI: https://doi.org/10.1090/S0002-9947-1977-0468710-X
MathSciNet review: 0468710
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Abstract: Let $\mathcal {L}$ be a complete projective logic. Then $\mathcal {L}$ 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 $\theta$-bilinear symmetric form on V, $\theta$ 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 $\theta$ is continuous, and if the dimension of $\mathcal {L}$ is properly restricted, then $\mathcal {L}$ is just one of the standard Hilbert space logics. Here we also assume $\mathcal {L}$ is a complete projective logic. Then if every $\theta$-fixed element of D is in the center of D and can be written as $\pm d\theta (d)$, some $d \in D$, and if the dimension of $\mathcal {L}$ is properly restricted, we show that $\mathcal {L}$ 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 $\theta$. Another is a purely lattice theoretic characterization of the lattice of closed subspaces of separable complex Hilbert space.


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Keywords: Complemented modular lattice, projective geometry, point, division ring, center of the ring, Archimedean field, non-Archimedean field, involutive antiautomorphism, <IMG WIDTH="16" HEIGHT="21" ALIGN="BOTTOM" BORDER="0" SRC="images/img2.gif" ALT="$\theta$">-bilinear symmetric form, complete projective logic, Hilbertian space, state, normalizable, probabilistic complete projective logic, infinite dimensional space, orthonormal sequence, standard quantum logics, separable Hilbert space
Article copyright: © Copyright 1977 American Mathematical Society