  A certain generalization of the mathematical formalism of quantum mechanics
beyond operator algebras is considered. The approach is based on the concept of
conditional probability and the interpretation of the Lueders - von Neumann
quantum measurement as a probability conditionalization rule. A major result
shows that the operator algebras must be replaced by order-unit spaces with
some specific properties in the generalized approach, and it is analyzed under
which conditions these order-unit spaces become Jordan algebras. An application
of this result provides a characterization of the projection lattices in
operator algebras.
