The Mackey-Gleason Problem

Abstract

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: "(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 ?{A) has a unique extension to a bounded linear functional on A .