(<Emphasis Type="Italic">o</Emphasis>)-Topology in *-algebras of locally measurable operators

We consider the topology t( M ) t\left( \mathcal{M} \right) of convergence locally in measure in the *-algebra LS( M ) LS\left( \mathcal{M} \right) of all locally measurable operators affiliated to the von Neumann algebra M \mathcal{M} . We prove that t( M ) t\left( \mathcal{M} \right) coincides with the (o)-topology in LS_h( M ) = { T ? LS( M ):T* = T } L{S_h}\left( \mathcal{M} \right) = \left\{ {T \in LS\left( \mathcal{M} \right):T* = T} \right\} if and only if the algebra M \mathcal{M} is σ-finite and is of finite type. We also establish relations between t( M ) t\left( \mathcal{M} \right) and various topologies generated by a faithful normal semifinite trace on M \mathcal{M} .