Generalized effect algebras of positive operators densely defined on Hilbert spaces
Effect algebras have been introduced by \textit{D. J. Foulis} and \textit{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.
scientific article; zbMATH DE number 5900903