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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A topology on quantum logics

Authors: Sylvia Pulmannová and Zdena Riečanová
Journal: Proc. Amer. Math. Soc. 106 (1989), 891-897
MSC: Primary 81B10; Secondary 06C15, 54A99
MathSciNet review: 967488
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Abstract: A uniform topology $ {\tau _M}$ induced by a set $ M$ of finite measures on a quantum logic $ L$ is studied. If $ m$ is a valuation on $ L$, the topology $ {\tau _m}$ induced by $ \{ m\} $ is equivalent to the topology induced by the pseudometric $ \rho (a,b) = m(a\Delta b)$. If the set $ M$ of measures is large enough, the topology $ {\tau _M}$ reflects in some sense the structure of $ L$: if $ L$ is a continuous geometry and the measures are totally additive, $ {\tau _M}$ is weaker than the order topology $ {\tau _o}$ on $ L$. If $ L$ is atomic, $ {\tau _M}$ is stronger than $ {\tau _o}$. On a separable Hilbert space logic, $ {\tau _M}$ coincides with the discrete topology. Some cases are found in which $ {\tau _M} = {\tau _o}$.

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Keywords: Orthomodular lattice, quantum logic, uniform topology generated by measures
Article copyright: © Copyright 1989 American Mathematical Society

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