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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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