Convergences in \(W^ *\)-algebras.

*(English)*Zbl 0612.46060Convergences 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).

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 |

##### Keywords:

lattice of projections; closely on large sets; nearly everywhere; almost uniformly; quasi-uniformly; sequence of observables; faithful normal state
Full Text:
DOI

**OpenURL**

##### References:

[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 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.