  Starting from an abstract setting for the Lueders - von Neumann quantum
measurement process and its interpretation as a probability conditionalization
rule in a non-Boolean event structure, the author derived a certain
generalization of operator algebras in a preceding paper. This is an order-unit
space with some specific properties. It becomes a Jordan operator algebra under
a certain set of additional conditions, but does not own a multiplication
operation in the most general case. A major objective of the present paper is
the search for such examples of the structure mentioned above that do not stem
from Jordan operator algebras; first natural candidates are matrix algebras
over the octonions and other nonassociative rings. Therefore, the case when a
nonassociative commutative multiplication exists is studied without assuming
that it satisfies the Jordan condition. The characteristics of the resulting
algebra are analyzed. This includes the uniqueness of the spectral resolution
as well as a criterion for its existence, subalgebras that are Jordan algebras,
associative subalgebras, and more different levels of compatibility than
occurring in standard quantum mechanics. However, the paper cannot provide the
desired example, but contribute to the search by the identification of some
typical differences between the potential examples and the Jordan operator
algebras and by negative results concerning some first natural candidates. The
possibility that no such example exists cannot be ruled out. However, this
would result in an unexpected new characterization of Jordan operator algebras,
which would have a significant impact on quantum axiomatics since some
customary axioms (e.g., powerassociativity or the sum postulate for
observables) might turn out to be redundant then.
