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A uniform Kadec-Klee property for symmetric operator spaces

Published online by Cambridge University Press:  24 October 2008

P. G. Dodds ,
T. K. Dodds ,
P. N. Dowling ,
C. J. Lennard  and
F. A. Sukochev
P. G. Dodds
Affiliation:
Information Science and Technology, The Flinders University of South Australia, GPO Box 2100, Adelaide, SA 5001, Australia
T. K. Dodds
Affiliation:
Information Science and Technology, The Flinders University of South Australia, GPO Box 2100, Adelaide, SA 5001, Australia
P. N. Dowling
Affiliation:
Mathematics and Statistics, Miami University, Oxford, OH 45056, USA
C. J. Lennard
Affiliation:
Mathematics and Statistics, University of Pittsburgh, Pittsburgh, PA 15260, USA
F. A. Sukochev
Affiliation:
Department of Mathematics, Tashkent State University, Vuzgorodok, 700095, Tashkent, Uzbekistan
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Abstract

We show that if a rearrangement invariant Banach function space E on the positive semi-axis satisfies a non-trivial lower q-estimate with constant 1 then the corresponding space E(M) of τ-measurable operators, affiliated with an arbitrary semi-finite von Neumann algebra M equipped with a distinguished faithful, normal, semi-finite trace τ, has the uniform Kadec-Klee property for the topology of local convergence in measure. In particular, the Lorentz function spaces Lq, p and the Lorentz-Schatten classes Cg, p have the UKK property for convergence locally in measure and for the weak-operator topology, respectively. As a partial converse, we show that if E has the UKK property with respect to local convergence in measure then E must satisfy some non-trivial lower q-estimate. We also prove a uniform Kadec-Klee result for local convergence in any Banach lattice satisfying a lower q-estimate.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 118 , Issue 3 , November 1995 , pp. 487 - 502
Copyright
Copyright © Cambridge Philosophical Society 1995

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