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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.

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