Spectra and variance of quantum random variables

We study essentially bounded quantum random variables and show that the Gelfand spectrum of such a quantum random variable ψ coincides with the hypoconvex hull of the essential range of ψ. Moreover, a notion of operator-valued variance is introduced, leading to a formulation of the moment problem in the context of quantum probability spaces in terms of operator-theoretic properties involving semi-invariant subspaces and spectral theory. As an application of quantum variance, new measures of random and inherent quantum noise are introduced for measurements of quantum systems, modifying some recent ideas of Polterovich [17].