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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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[1] Batty, C.J.K, The strong law of large numbers for states and traces of a W∗-algebra, Z. wahrsch. verw. gebiete, 48, 177-191, (1979) · Zbl 0395.60033
[2] Goldstein, M.S, Theorems on almost everywhere convergence in von Neumann algebras, J. operator theory, 6, 233-311, (1981), [in Russian] · Zbl 0488.46053
[3] Halmos, P.R, Two subspaces, Trans. amer. math. soc., 144, 381-389, (1969) · Zbl 0187.05503
[4] Lance, C, Almost uniform convergence in operaor algebras, (), 136-142
[5] Petz, D, Quasi-uniform ergodic theorems in von Neumann algebras, Bull. London math. soc., 16, 151-156, (1984) · Zbl 0535.46042
[6] Stratila, S; Zsido, L, Lectures on von Neumann algebras, (1979), Abacus Press Tunbridge Wells, Kent England · Zbl 0391.46048
[7] Takesaki, M, Theory of operator algebras. I, (1979), Springer-Verlag Berlin/Heidelberg/New York · Zbl 0990.46034
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