Comparison of topologies on ⁎-algebras of locally measurable operators

We consider the local measure topology \({t(\mathcal{M})}\) on the ⁎-algebra \({LS(\mathcal{M})}\) of all locally measurable operators and on the ⁎-algebra \({S(\mathcal{M},\tau)}\) of all τ-measurable operators affiliated with a von Neumann algebra \({\mathcal{M}}\). If τ is a semifinite but not a finite trace on \({\mathcal{M},}\) then one can consider the τ-local measure topology tτ l and the weak τ-local measure topology twτ l. We study relationships between the topology \({t(\mathcal{M})}\) and the topologies tτ l, twτ l, and the (o)-topology \({t_o(\mathcal{M})}\) on \({LS_h(\mathcal{M})=\{T\in LS(\mathcal{M}): T^\ast=T\}}\). We find that the topologies \({t(\mathcal{M})}\) and tτ l (resp. \({t(\mathcal{M})}\) and twτ l) coincide on \({S(\mathcal{M},\tau)}\) if and only if \({\mathcal{M}}\) is finite, and \({t(\mathcal{M})=t_o(\mathcal{M})}\) on \({LS_h(\mathcal{M})}\) holds if and only if \({\mathcal{M}}\) is a σ-finite and finite. Moreover, it turns out that the topology tτl (resp. twτ l) coincides with the (o)-topology on \({S_h(\mathcal{M},\tau)}\) only for finite traces. We give necessary and sufficient conditions for the topology \({t(\mathcal{M})}\) to be locally convex (resp., normable). We show that (o)-convergence of sequences in \({LS_h(\mathcal{M})}\) and convergence in the topology \({t(\mathcal{M})}\) coincide if and only if the algebra \({\mathcal{M}}\) is an atomic and finite algebra.