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A topology on quantum logics. (English) Zbl 0679.06004
A quantum logic, L, is a \(\sigma\)-complete orthomodular lattice. A measure on L is a map m: \(L\to R^+\) so that m(\(\bigvee_{i\in N}a_ i)=\sum_{i\in N}m(a_ i)\), for any sequence, \((a_ i)_{i\in N}\), of pairwise orthogonal elements of L. The interplay between the supply of measures on a logic L and its lattice theoretic structure is often of importance. In the paper under review the authors initiate a study of this interplay by topological means. A set of measures, M, induces a pseudometric topology, \(\tau_ M\), on the elements of L. This topology is examined and compared to various topologies induced by the order structure of L. In the case of the logic of closed subspaces of a separable Hilbert space, and of a separating (i.e. for all \(a\in L\), there exists \(m\in M\) with m(a)\(\neq 0)\) set of measures, \(\tau_ M\) is just the discrete topology.
Reviewer: M.Roddy

MSC:
06C15 Complemented lattices, orthocomplemented lattices and posets
06B30 Topological lattices
03G12 Quantum logic
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
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