The Mackey-Gleason Problem

Let A be a von Neumann algebra with no direct summand of Type <!-- MATH ${{\text{I}}_2}$ --> , and let <!-- MATH $\mathcal{P}(A)$ --> be its lattice of projections. Let X be a Banach space. Let <!-- MATH $m:\mathcal{P}(A) \to X$ --> be a bounded function such that <!-- MATH $m(p + q) = m(p) + m(q)$ --> whenever p and q are orthogonal projections. The main theorem states that m has a unique extension to a bounded linear operator from A to X. In particular, each bounded complex-valued finitely additive quantum measure on <!-- MATH $\mathcal{P}(A)$ --> has a unique extension to a bounded linear functional on A.