Local convergence in measure on semifinite von Neumann algebras

Suppose that ? is a von Neumann algebra of operators on a Hilbert space $\mathcal{H}$ and τ is a faithful normal semifinite trace on ?. The set of all τ-measurable operators with the topology t _τ of convergence in measure is a topological *-algebra. The topologies of τ-local and weakly τ-local convergence in measure are obtained by localizing t _τ and are denoted by t _τ1 and t _wτ1, respectively. The set with any of these topologies is a topological vector space. The continuity of certain operations and the closedness of certain classes of operators in with respect to the topologies t _τ1 and t _wτ1 are proved. S.M. Nikol’skii’s theorem (1943) is extended from the algebra $\mathcal{B}(\mathcal{H})$ to semifinite von Neumann algebras. The following theorem is proved: For a von Neumann algebra ? with a faithful normal semifinite trace τ, the following conditions are equivalent: (i) the algebra ? is finite; (ii) t _wτ1 = t _τ1; (iii) the multiplication is jointly t _τ1-continuous from to ; (iv) the multiplication is jointly t _τ1-continuous from to ; (v) the involution is t _τ1-continuous from to .