The Mackey-Gleason problem

Bunce, L. J.; Wright, J. D. Maitland

doi:10.1090/S0273-0979-1992-00274-4

The Mackey-Gleason problem
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by L. J. Bunce and J. D. Maitland Wright PDF

Bull. Amer. Math. Soc. 26 (1992), 288-293 Request permission

Abstract:

Let A be a von Neumann algebra with no direct summand of Type ${{\text {I}}_2}$, and let $\mathcal {P}(A)$ be its lattice of projections. Let X be a Banach space. Let $m:\mathcal {P}(A) \to X$ 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 $\mathcal {P}(A)$ has a unique extension to a bounded linear functional on A.

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