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

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 \( \mathcal{M} \).


Local Convergence Partial Isometry Strong Operator Topology Closed Linear Operator Complete Topological 
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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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