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Probability measures on projections in von Neumann algebras. (English) Zbl 0718.46046

A state \(\phi\) on a von Neumann algebra A is a positive linear-functional on A with \(\phi (1)=1\). The restriction of \(\phi\) to the set of projections on A is a finitely additive probability measure. The aim of this note is to give precise and complete arguments for proving the following facts:
If A is a von Neumann algebra without direct summand of type \(I_ 2\), and let \(\mu\) be a finitely additive probability measure on the complete orthomodular lattice P(A) of projections in A. Then every \(\mu\) can be extended to a state \({\hat \mu}\) on A (i.e. \({\hat \mu}\) is a positive linear functional with \(\| {\hat \mu}\| =\mu (1)=1)\), moreover \(| \mu (e)-\mu (f)| \leq \| e-f\| \quad (e,f\in P(A)).\)
The following three statements are equivalent:
(\(\alpha\)) \(\mu\) is completely additive.
(\(\beta\)) \(\mu\) is \(\sigma\)-finite and has the support.
(\(\gamma\)) \(\mu\) can be extended to a normal state \({\hat \mu}\) on A.
In order to prove the above facts, the author begins to describe basic notions of algebra operators and orthomodular posets of projections, continuity of probability measures on projections, and linear extensions of probability measure on projections through the three chapters concisely and systematically. Summing up of these results he proved the above final results.
Reviewer: J.C.Rho (Seoul)

MSC:

46L51 Noncommutative measure and integration
46L53 Noncommutative probability and statistics
46L54 Free probability and free operator algebras
46L60 Applications of selfadjoint operator algebras to physics
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