$(o)$-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 $\mathcal{M}$