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Zbl 1099.81010
Buhagiar, David; Chetcuti, Emanuel
On complete-cocomplete subspaces of an inner product space. (English)
[J] Appl. Math., Praha 50, No. 2, 103-114 (2005). ISSN 0862-7940; ISSN 1572-9109/e

Summary: In this note we give a measure-theoretic criterion for the completeness of an inner product space. We show that an inner product space $S$ is complete if and only if there exists a $\sigma $-additive state on $C(S)$, the orthomodular poset of complete-cocomplete subspaces of $S$. We then consider the problem of whether every state on $E(S)$, the class of splitting subspaces of $S$, can be extended to a Hilbertian state on $E(\bar S)$; we show that for the dense hyperplane $S$ (of a separable Hilbert space) constructed by {\it P. Pták} and {\it H. Weber} [Proc. Am. Math. Soc. 129, 2111-2117 (2001; Zbl 0968.03077)], every state on $E(S)$ is a restriction of a state on $E(\bar S)$.

MSC 2000:: *81P10 Logical foundations of quantum mechanics
03G12 Quantum logic

Keywords: Hilbert space; inner product space; orthogonally closed subspace; complete and cocomplete subspaces; finitely and $\sigma $-additive state

Citations: Zbl 0968.03077

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