The Mackey-Gleason-Bunce-Wright problem for vector-valued measures on projections in a JBW${}^{\ast }$-algebra

Abstract

Let $\mathcal{P}(\mathfrak{J})$ denote the lattice of projections of a JBW${}^{\ast }$-algebra $\mathfrak{J}$, and let $X$ be a Banach space. A bounded finitely additive $X$-valued measure on $\mathcal{P}(\mathfrak{J})$ is a mapping $\hat{I}¼:\mathcal{P}(\mathfrak{J})\to X$ satisfying: $(a)$ $\hat{I}¼(p+q)=\hat{I}¼(p)+\hat{I}¼(q)$, whenever $p\circ q=0$ in $\mathcal{P}(\mathfrak{J})$, $(b)$ $\sup \{\Vert \hat{I}¼(p)\Vert :p\in \mathcal{P}(\mathfrak{J})\}<\infty$. In this paper we establish a Mackey-Gleason-Bunce-Wright theorem by showing that if $\mathfrak{J}$ contains no type $I_2$ direct summand, every bounded finitely additive measure $\hat{I}¼:\mathcal{P}(\mathfrak{J})\to X$ admits an extension to a bounded linear operator from $\mathfrak{J}$ to $X$. This solves a long-standing open conjecture.