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Transactions of the American Mathematical Society

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



Quantum logic and the locally convex spaces

Author: W. John Wilbur
Journal: Trans. Amer. Math. Soc. 207 (1975), 343-360
MSC: Primary 46A05; Secondary 81.02
MathSciNet review: 0367607
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Abstract: An important theorem of Kakutani and Mackey characterizes an infinite dimensional real (complex) Hilbert space as an infinite dimensional real (complex) Banach space whose lattice of closed subspaces admits an orthocomplementation. This result, also valid for quaternionic spaces, has proved useful as a justification for the unique role of Hilbert space in quantum theory. With a like application in mind, we present in the present paper a number of characterizations of real and complex Hilbert space in the class of locally convex spaces. One of these is an extension of the Kakutani-Mackey result from the infinite dimensional Banach spaces to the class of all infinite dimensional complete Mackey spaces. The implications for the foundations of quantum theory are discussed.

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Keywords: Quantum logic, projective logic, state, question lattice, lattice of closed subspaces, orthocomplementation, $ \theta $-bilinear symmetric form, continuous automorphism, discontinuous automorphism, $ \theta $-bilinear space, Hilbertian space, inner product space, Hilbert space, metrizable space, Mackey space
Article copyright: © Copyright 1975 American Mathematical Society