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Generalized effect algebras of positive operators densely defined on Hilbert spaces. (English) Zbl 1237.81009
Effect algebras have been introduced by D. J. Foulis and M. K. Bennett [“Effect algebras and unsharp quantum logics”, Found. Phys. 24, No. 10, 1331–1352 (1994; Zbl 1213.06004)] for modelling unsharp measurement in a quantum mechanical system.
The authors of the paper under review consider examples of sets of positive linear operators defined on a dense linear subspace \(D\) in a (complex) Hilbert space \(\mathcal H\). Some of these operators may have a physical meaning in quantum mechanics. It is proved that the set of all positive linear operators with fixed such \(D\) and \(\mathcal H\) form a generalized effect algebra with respect to the usual addition of operators. Some sub-algebras are also mentioned. Moreover, on a set of all positive linear operators densely defined in an infinite dimensional complex Hilbert space, the partial binary operation is defined making this set a generalized effect algebra.

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