Local convergence in measure on semifinite von Neumann algebras

Suppose that ℳ is a von Neumann algebra of operators on a Hilbert space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal{H}$$ \end{document} and τ is a faithful normal semifinite trace on ℳ. The set [graphic not available: see fulltext] of all τ-measurable operators with the topology tτ of convergence in measure is a topological *-algebra. The topologies of τ-local and weakly τ-local convergence in measure are obtained by localizing tτ and are denoted by tτ1 and twτ1, respectively. The set [graphic not available: see fulltext] with any of these topologies is a topological vector space. The continuity of certain operations and the closedness of certain classes of operators in [graphic not available: see fulltext] with respect to the topologies tτ1 and twτ1 are proved. S.M. Nikol’skii’s theorem (1943) is extended from the algebra \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal{B}(\mathcal{H})$$ \end{document} 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) twτ1 = tτ1; (iii) the multiplication is jointly tτ1-continuous from[graphic not available: see fulltext] to [graphic not available: see fulltext]; (iv) the multiplication is jointly tτ1-continuous from[graphic not available: see fulltext] to [graphic not available: see fulltext]; (v) the involution is tτ1-continuous from[graphic not available: see fulltext] to [graphic not available: see fulltext].