Why do the quantum observables form a Jordan operator algebra? (Q702983)

This paper is a continuation of a previous work by the same author [J. Phys. A, Math. Gen. 34, 6031--6042 (2001; Zbl 0981.81015)], where a non-Boolean extension of classical probabilities was presented in order to apply it in the quantum measurement process. Here the author shows how to derive the distributive nonassociative multiplication operation of a Jordan algebra of operators associated to quantum observables, from certain properties of the conditional probabilities and the observables in this new framework, which has a clear statistical interpretation.