is uniformly Kadec-Klee

Author:
Chris Lennard

Journal:
Proc. Amer. Math. Soc. **109** (1990), 71-77

MSC:
Primary 46B20; Secondary 47D25

DOI:
https://doi.org/10.1090/S0002-9939-1990-0943795-4

MathSciNet review:
943795

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Abstract: A dual Banach space is Kadec-Klee in the weak * topology if weak * and norm convergence of sequences coincide in the unit sphere of . We shall consider a stronger, uniform version of this property. A dual Banach space is uniformly Kadec-Klee in the weak * topology (UKK*) if for each we can find a in such that every weak -compact, convex subset of the unit ball of whose measure of norm compactness exceeds must meet the -ball of . We show in this paper that , the space of trace class operators on an arbitrary infinite-dimensional Hilbert space is UKK*. Consequently has weak -normal structure. This answers affirmatively a question of A. T. Lau and P. F. Mah. From this it follows that has the weak -fixed point property.

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DOI:
https://doi.org/10.1090/S0002-9939-1990-0943795-4

Article copyright:
© Copyright 1990
American Mathematical Society