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A limit in probability in a \(W^*\)-algebra is unique. (English) Zbl 0821.46081
In the noncommutative probability setting, a von Neumann algebra \({\mathbf M}\subset B(H)\) with a faithful normal state \(\omega\) is treated as the generalization of \(L^ \infty(\Omega, S, P)\) over a probability space \((\Omega, S, P)\). If \(x_ n\), \(x\) are (not necessarily bounded) operators on \(H\) then \(x_ n\to x\) by definition if there exists a sequence \((e_ n)\subset {\mathbf M}\) of projections such that \(\omega(e_ n)\to 1\) and \(\|(x_ n- x)e_ n\|\to 0\). The main result says that a sequence of selfadjoint operators affiliated with \({\mathbf M}\) may have at most one Segal measurable limit. The author also shows that a sequence converging in probability contains a subsequence converging almost uniformly.

46L51 Noncommutative measure and integration
46L53 Noncommutative probability and statistics
46L54 Free probability and free operator algebras
Full Text: DOI
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