The Mackey-Gleason Problem

Abstract

Let $A$ be a von Neumann algebra with no direct summand of Type $\roman I_2$, and let $\scr P(A)$ be its lattice of projections. Let $X$ be a Banach space. Let $m\:\scr P(A)\to X$ 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 $\scr P(A)$ has a unique extension to a bounded linear functional on $A$.