The Mackey–Gleason–Bunce–Wright problem for vector-valued measures on projections in a JBW*-algebra

Let P(J) P ( J ) denote the lattice of projections of a JBW* JBW ∗ -algebra J J , and let X be a Banach space. A bounded finitely additive X -valued measure on P(J) P ( J ) is a map μ : P(J) → X μ : P ( J ) → X satisfying ( a ) μ (p +q) = μ (p) + μ (q) μ ( p + q ) = μ ( p ) + μ ( q ) , whenever p ∘ q = 0 p ∘ q = 0 in P(J) P ( J ) , ( b ) \ μ (p) : \, p ∈ P(J)\ < ∞ sup { ‖ μ ( p ) ‖ : p ∈ P ( J ) } < ∞ . In this paper we establish a Mackey–Gleason–Bunce–Wright theorem by showing that if J J contains no type <jats:inline-formula