THE MACKEY-GLEASON PROBLEM

Let A be a von Neumann algebra with no direct summand of Type I2, and let P(A) be its lattice of projections. Let X be a Banach space. Let m: P(A) --> X be a bounded function such that 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 P(A) has a unique extension to a bounded linear functional on A .