Local convergence in measure on semifinite von Neumann algebras

Suppose that ℳ is a von Neumann algebra of operators on a Hilbert space $$\mathcal{H}$$ and τ is a faithful normal semifinite trace on ℳ. 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 ℳ with a faithful normal semifinite trace τ, the following conditions are equivalent: (i) the algebra ℳ 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 .

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