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

We consider the topology t( M ) tleft( mathcal{M} right) of convergence locally in measure in the *-algebra LS( M ) LSleft( mathcal{M} right) of all locally measurable operators affiliated to the von Neumann algebra M mathcal{M} . We prove that t( M ) tleft( 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 LSleft( 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 ) tleft( mathcal{M} right) and various topologies generated by a faithful normal semifinite trace on M mathcal{M} .