: Nonassociative operator observable algebras, like those that occur in the presence of small magnetic monopole charges and non-geometric backgrounds in string theory, cannot be represented as linear operators on a Hilbert space, creating difficulties in extracting physics from these systems. Starting with a more robust definition of an eigenstate and using inequalities between moments following from positivity of the state to find the spectra of several operators. This is done without making any reference to wave functions or density matrices, and can thus be applied to nonassociative operator algebras as well. From this, we find a new bound on the magnetic charge of the muon (g ≤ 4.7·10−18Am = 1.4·10−9 gDirac).