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The Mackey-Gleason problem. (English) Zbl 0759.46054
Let $P(A)$ be the lattice of projections in a von Neumann algebra $A$ and let $X$ be a Banach space. A function $\mu: P(A)\to X$ is called a finitely additive $X$-valued measure on $P(A)$ if $\mu(e+f)=\mu(e)+\mu(f)$ whenever $ef=0$, and $\sup\{\Vert\mu(e)\Vert$: $e\in P(A)\}<\infty$. The main result of the paper is the following theorem. Theorem. Let $A$ be a von Neumann algebra with no direct summand of type $I\sb 2$. Then, for any Banach space $X$, each finitely additive $X$- valued measure on $P(A)$ has a unique extension to a bounded linear operator from $A$ to $X$. In particular, each bounded complex-valued finitely additive quantum measure on $P(A)$ has a unique extension to a bounded linear functional on $A$.

MSC:
46L51Noncommutative measure and integration
46L53Noncommutative probability and statistics
46L54Free probability and free operator algebras
46G10Vector-valued measures and integration
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