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A uniform Kadec-Klee property for symmetric operator spaces

Peter G. DoddsInformation Science and Technology, The Flinders University of South Australia, GPO Box 2100, Adelaide, SA 5001, AustraliaTheresa K.-Y. DoddsInformation Science and Technology, The Flinders University of South Australia, GPO Box 2100, Adelaide, SA 5001, AustraliaPatrick N. DowlingMathematics and Statistics, Miami University, Oxford, OH 45056, USAChris LennardMathematics and Statistics, University of Pittsburgh, Pittsburgh, PA 15260, USAFedor SukochevDepartment of Mathematics, Tashkent State University, Vuzgorodok, 700095, Tashkent, Uzbekistan

Mathematical Proceedings of the Cambridge Philosophical Societyjournal1995en

ABI

Annotatsiya

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 L q, p and the Lorentz-Schatten classes C g, 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.

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Mavzular

Advanced Banach Space Theory Advanced Operator Algebra Research Holomorphic and Operator Theory

Identifikatorlar

DOI: 10.1017/s0305004100073813

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