## On characterizing the standard quantum logics

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- by W. John Wilbur
- Trans. Amer. Math. Soc.
**233**(1977), 265-282 - DOI: https://doi.org/10.1090/S0002-9947-1977-0468710-X
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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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## Bibliographic Information

- © Copyright 1977 American Mathematical Society
- 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