The sum of observables on a $\sigma$-distributive lattice effect algebra

Abstract

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.