Completeness of inner product spaces and quantum logic of splitting subspaces |
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Authors: | Anatolij Dvurečenskij |
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Institution: | (1) Mathematical Institute, Slovak Academy of Sciences, Obrancov mieru 49, CS-814 73 Bratislava, Czechoslovakia |
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Abstract: | We show that an inner product space S is complete whenever its system E(S) of all splitting subspaces, i.e., of all subspaces E for which E E
=S holds, is a quantum logic, that is, an orthocomplemented orthomodular -orthoposet. It is well known that the quantum logic is an important axiomatic model of quantum mechanics. This generalizes the result of G. Cattaneo and G. Marino (Lett. Math. Phys.
11, 15–20 (1986)) who required that E(S) be a lattice. Moreover, the conditions are weakened to show that S is complete whenever E(S) contains the join of any sequence of one-dimentional orthogonal subspaces. |
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Keywords: | 46C10 03G12 81B10 |
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