J Austral Math Soc Ser A 59 pp343--352, 1995.

On the Uniform Kadec-Klee Property With Respect to Convergence in Measure

F. A. Sukochev

Abstract

Let E(0, ¥) be a separable symmetric function space, let M be a semifinite von Neumann algebra with normal faithful semifinite trace m, and let E(M, m) be the symmetric operator space with E(0, ¥). If E(0, ¥) has the uniform Kadec-Klee property with respect to convergence in measure then E(M, m) also has this property. In particular, if L_F (0, ¥)(L _f (0, ¥)) is a separable Orlicz (Lorentz) space then L_F (M, m)(L _f (M, m)) has the uniform Kadec-Klee property with respect to convergence in measure. It is established also that E(0, ¥) has the uniform Kadec-Klee property with respect to convergence in measure on sets of finite measure if and only if the norm of E(0, ¥) satisfies G. Birkhoff's condition of uniform monotonicity.

1991 AMS Subject Classification: primary 46B20; secondary 46E30, 46L50

Browse the article

Authors

F. A. Sukochev

Department of Mathematics and Statistics, The Flinders University, G. P. O. Box 2100, Adelaide, SA 5001, Australia.

mailto:sukochev@ist.flinders.edu.au