The Mackey-Gleason problem

Let A be a von Neumann algebra with no direct summand of Type I 2 {{I}_2} , and let P ( A ) P(A) be its lattice of projections. Let X be a Banach space. Let m : P ( A ) → X m:P(A) → X be a bounded function such that m ( p + q ) = m ( p ) + m ( q ) 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 ) P(A) has a unique extension to a bounded linear functional on A .