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 \( \mathcal{M} \).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 11, pp. 1531–1540, November, 2009.
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Muratov, M.A., Chilin, V.I. (o)-Topology in *-algebras of locally measurable operators. Ukr Math J 61, 1798–1808 (2009). https://doi.org/10.1007/s11253-010-0313-y
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DOI: https://doi.org/10.1007/s11253-010-0313-y