  The extended semantic realism (ESR) model proposes a new theoretical
perspective which embodies the mathematical formalism of standard (Hilbert
space) quantum mechanics (QM) into a noncontextual framework, reinterpreting
quantum probabilities as conditional instead of absolute. We provide in this
review an overall view on the present status of our research on this topic. We
attain in a new, shortened way a mathematical representation of the generalized
observables introduced by the ESR model and a generalization of the projection
postulate of elementary QM. Basing on these results we prove that the
Bell-Clauser-Horne-Shimony-Holt (BCHSH) inequality, a modified BCHSH inequality
and quantum predictions hold together in the ESR model because they refer to
different parts of the picture of the physical world supplied by the model.
Then we show that a new mathematical representation of mixtures must be
introduced in the ESR model which does not coincide with the standard
representation in QM and avoids some deep problems that arise from the
representation of mixtures provided by QM. Finally we get a nontrivial
generalization of the Luders postulate, which is justified in a special case by
introducing a reasonable physical assumption on the evolution of the compound
system made up of the measured system and the measuring apparatus.
