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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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On characterizing the standard quantum logics
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by W. John Wilbur PDF
Trans. Amer. Math. Soc. 233 (1977), 265-282 Request permission

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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Additional 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