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On the uniform Kadec-Klee property with respect to convergence in measure

Part of: Normed linear spaces and Banach spaces; Banach lattices Linear function spaces and their duals

Published online by Cambridge University Press:  09 April 2009

F. A. Sukochev
F. A. Sukochev
Affiliation:
Department of Mathematics and StatisticsThe Flinders UniversityG.P.O. Box 2100Adelaide, SA 5001Australia e-mail: sukochev@ist.flinders.edu.au
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Abstract

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Let E(0, ∞) be a separable symmetric function space, let M be a semifinite von Neumann algebra with normal faithful semifinite trace μ, and let E(M, μ) be the symmetric operator space associated with E(0, ∞). If E(0, ∞) has the uniform Kadec-Klee property with respect to convergence in measure then E(M, μ) also has this property. In particular, if LΦ(0, ∞) (ϕ(0, ∞)) is a separable Orlicz (Lorentz) space then LΦ(M, μ) (Λϕ (M, μ)) 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.

MSC classification

Secondary: 46B20: Geometry and structure of normed linear spaces 46E30: Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 59 , Issue 3 , December 1995 , pp. 343 - 352
Copyright
Copyright © Australian Mathematical Society 1995

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