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Ukrainian Mathematical Journal

, Volume 61, Issue 11, pp 1798–1808 | Cite as

(o)-Topology in *-algebras of locally measurable operators

  • M. A. Muratov
  • V. I. Chilin
Article

We consider the topology \( t\left( \mathcal{M} \right) \) of convergence locally in measure in the *-algebra \( LS\left( \mathcal{M} \right) \) of all locally measurable operators affiliated to the von Neumann algebra \( \mathcal{M} \). We prove that \( t\left( \mathcal{M} \right) \) coincides with the (o)-topology in \( L{S_h}\left( \mathcal{M} \right) = \left\{ {T \in LS\left( \mathcal{M} \right):T* = T} \right\} \) if and only if the algebra \( \mathcal{M} \) is σ-finite and is of finite type. We also establish relations between \( t\left( \mathcal{M} \right) \) and various topologies generated by a faithful normal semifinite trace on \( \mathcal{M} \).

Keywords

Local Convergence Partial Isometry Strong Operator Topology Closed Linear Operator Complete Topological 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media, Inc. 2009

Authors and Affiliations

  • M. A. Muratov
    • 1
  • V. I. Chilin
    • 2
  1. 1.Tavrida National UniversitySimferopolUkraine
  2. 2.Uzbekistan National UniversityTashkentUzbekistan

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