A limit in probability in a \(W^*\)-algebra is unique.

*(English)*Zbl 0821.46081In 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.

##### MSC:

46L51 | Noncommutative measure and integration |

46L53 | Noncommutative probability and statistics |

46L54 | Free probability and free operator algebras |

##### Keywords:

noncommutative probability; von Neumann algebra; faithful normal state; Segal measurable; sequence converging in probability contains a subsequence converging almost uniformly
PDF
BibTeX
XML
Cite

\textit{A. Paszkiewicz}, J. Funct. Anal. 90, No. 2, 429--444 (1990; Zbl 0821.46081)

Full Text:
DOI

**OpenURL**

##### References:

[1] | Batty, C. J. K.: The strong law of large numbers for states and traces of a w\ast-algebra. Z. wahrsch. Verw. gebiete 48, 177-191 (1979) · Zbl 0395.60033 |

[2] | E. Ignaczak, Convergence of subsequences in W\ast-algebra, to appear. |

[3] | Jajte, R.: Strong limit theorems in non-commutative probability. Lecture notes math. 1110 (1985) · Zbl 0554.46033 |

[4] | Nelson, E.: Notes on non-commutative integration. J. funct. Anal. 15, 103-116 (1974) · Zbl 0292.46030 |

[5] | Paszkiewicz, A.: Convergences in w\ast-algebras. J. funct. Anal. 69, 143-154 (1986) · Zbl 0612.46060 |

[6] | A. Paszkiewicz, An almost uniform limit in a W\ast-algebra is unique, Demonstratio Math., to appear. · Zbl 0723.46043 |

[7] | A. Paszkiewicz, Convergences in W\ast-algebras–their strange behaviour and tools for their investigation, in ”Proceedings, Symposium on Quantum Probability and Applications, Rome, October ’86–July ’87,” in Lecture Notes in Math., to appear. |

[8] | Segal, I. E.: A non-commutative extension of abstract integration. Ann. math. 57, 401-457 (1953) · Zbl 0051.34201 |

[9] | Stratila, S.; Zsido, L.: Lectures on von Neumann algebras. (1979) |

[10] | Takesaki, M.: Theory of operator algebras, I. (1979) · Zbl 0436.46043 |

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.