2017
Том 69
№ 9

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

Abstract

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 $\mathcal{M}$.

English version (Springer): Ukrainian Mathematical Journal 61 (2009), no. 11, pp 1798-1808.

Citation Example: Chilin V. I., Muratov M. A. $(o)$-Topology in *-algebras of locally measurable operators // Ukr. Mat. Zh. - 2009. - 61, № 11. - pp. 1531-1540.

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