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Vector measures on orthocomplemented lattices. (English) Zbl 0819.46036
Let \(L\) be a relatively orthocomplemented lattice [G. Birkhoff, Lattice theory. 3rd ed., Providence, RI: AMS (1967; Zbl 0153.02501)]. If \(H\) is a Hilbert space, a map \(m\colon L\to H\) is said to be orthogonally scattered if \(m(a)\perp m(b)\) whenever \(a\perp b\). Denoting the lattice of orthogonal projections on \(H\) by \(L(H)\), the author proves the following representation theorem: For every \(H\)-valued bounded countably additive orthogonally scattered measure \(m\) on \(L\) there exists a \(\sigma\)-homomorphism \(\varphi\colon L\to L(H)\) and a vector \(x\) in \(H\) such that \(m(a)= \varphi(a)x\). This generalizes a result of P. Masani [Adv. Math. 2, 61–117 (1968; Zbl 0187.38705)].
The existence of such measures is also studied. For example, it is shown that there exists a nonzero countably additive orthogonally scattered measure from \(L(H)\) to \(L(K)\) if and only if \(\dim H\le \dim K\). Some applications to the quantum logic approach to axiomatic quantum mechanics are given.

46G10 Vector-valued measures and integration
06C15 Complemented lattices, orthocomplemented lattices and posets
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)