LOCAL CONVERGENCE IN MEASURE ON SEMIFINITE VON NEUMANN ALGEBRAS. III

. Let be a von Neumann algebra in a Hilbert space H and tau be a normal faithful semifinite trace on M. The set (M) over bar all tau-measurable operators with the topology t(tau) of convergence in measure is a topological *-algebra. The topologies of T-local and weak T-local convergence in measure are obtained by localizing of t(tau) and are denoted by t(tau)l and t(omega tau)l. respectively. The set (M) over bar with any of these topologies is a topological vector space. The continuity of some operations and closeness of certain classes of operators in (M) over bar with respect to the topologies t(tau)l and t(omega tau)l are proved. We consider the restrictions of the topologies t(tau)l and t(tau)l to the projection lattice and formulate an open problem. We obtain new characterizations of finiteness and sigma-finiteness of the algebra M by terms of t(tau)l and prove some convexity properties of the topology t(tau)l.