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Convergences in $$W^ *$$-algebras. (English) Zbl 0612.46060
Convergences closely on large sets (c.l.s.), nearly everywhere (n.e.), almost uniformly (a.u.) and quasi-uniformly (q.u.) for a sequence of observables from a $$W^*$$-algebra $${\mathfrak A}$$ with a faithful normal state $$\rho$$ are all of ”a.e.-type”. Namely, in the commutative case, where $${\mathfrak A}=L^{\infty}(\Omega,\mu)$$ and $$\rho (x)=\int xd\mu$$, each of these types of convergence is equivalent to the $$\mu$$-a.e. one in $$\Omega$$.
The main theorem of the paper states, in particular, that the q.u. convergence implies a.u. one, a.u. implies c.l.s., c.l.s. is equivalent to n.e. Moreover, they are all equivalent for a bounded sequence of operators and, in general, they are not equivalent for unbounded sequences. For convergence of subsequences, see the paper ”Convergences almost everywhere in $$W^*$$-algebras” by the author in Lect. Notes Math. 1136, 420-427 (1985; Zbl 0576.46047).

##### MSC:
 46L51 Noncommutative measure and integration 46L53 Noncommutative probability and statistics 46L54 Free probability and free operator algebras 46L10 General theory of von Neumann algebras 60A05 Axioms; other general questions in probability
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##### References:
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