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The sum of observables on a \(\sigma\)-distributive lattice effect algebra. (English) Zbl 1418.03181
Summary: Observables on quantum structures can be seen as generalizations of random variables on a measurable space \((\Omega , \mathcal {A})\) for the case when \(\mathcal {A}\) is not necessarily a Boolean algebra. The present paper investigates an extending of the usual pointwise sum of random variables onto the set of bounded observables on a \(\sigma\)-distributive lattice effect algebra \(E\). We describe conditions under which this operation, so-called sum \(x+y\) of observables \(x\), \(y\), preserves continuity of spectral resolutions of \(x\), \(y\). We show how the spectrum \(\sigma (x+y)\) depends on spectra \(\sigma (x)\), \(\sigma (y)\), and we provide a relation between the meager part \(x_m\) and the dense part \(x_d\) of an observable \(x\).
MSC:
03G12 Quantum logic
06C15 Complemented lattices, orthocomplemented lattices and posets
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
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