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Hilbert space effect-representations of effect algebras. (English) Zbl 1268.81014
Summary: In answer to open questions (posed in [J. Paseka and the first author, Found. Phys. 41, No. 10, 1634–1647 (2011; Zbl 1238.81009)]) we prove that an effect algebra has a Hilbert space effect-representation iff \(E\) possesses an ordering set of states. These are, up to isomorphism, all intervals and all their sub-effect algebras in the set of all positive linear operators on any Hilbert space \(\mathcal H\). Nevertheless, there are effect algebras \(E\), elements of which are linear operators in a Hilbert space, but \(E\) does not have such a representation.

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