$(o)$-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 \( \mathcal{M} \).
How to Cite
Muratov, M. A., and V. I. Chilin. “$(o)$-Topology in *-Algebras of Locally Measurable Operators”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, no. 11, Nov. 2009, pp. 1531-40, https://umj.imath.kiev.ua/index.php/umj/article/view/3119.
Research articles