On the problem of boundedness of a signed measure on projections of a von Neumann algebra

Let M be a von Neumann algebra and Mn be the set of all orthogonal projections in M. We call a mapping ηMn → C a signed measure on M if η is totally orthoadditive, that is, η(∑i ε{lunate} IPi) = ε{lunate}i ε{lunate} I η(Pi) for Pi ε{lunate} Mn, Pi⊥ Pj (i ≠ j). Here the condition of boundedness is usually required for the effective study and application of signed measures. So a natural problem of the existence of unbounded signed measures arises. In the present paper it is proved that any signed measure on the set of projections of a continuous von Neumann algebra is bounded. This fact is generalized also for vector-valued measures. © 1992