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$ \mathcal{C}_1$ is uniformly Kadec-Klee

Author: Chris Lennard
Journal: Proc. Amer. Math. Soc. 109 (1990), 71-77
MSC: Primary 46B20; Secondary 47D25
MathSciNet review: 943795
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Abstract: A dual Banach space $ X$ is Kadec-Klee in the weak * topology if weak * and norm convergence of sequences coincide in the unit sphere of $ X$. We shall consider a stronger, uniform version of this property. A dual Banach space $ X$ is uniformly Kadec-Klee in the weak * topology (UKK*) if for each $ \varepsilon > 0$ we can find a $ \delta $ in $ (0,1)$ such that every weak $ *$-compact, convex subset $ C$ of the unit ball of $ X$ whose measure of norm compactness exceeds $ \varepsilon $ must meet the $ (1 - \delta )$-ball of $ X$. We show in this paper that $ {C_1}(\mathcal{H})$, the space of trace class operators on an arbitrary infinite-dimensional Hilbert space $ \mathcal{H}$ is UKK*. Consequently $ {C_1}(\mathcal{H})$ has weak $ *$-normal structure. This answers affirmatively a question of A. T. Lau and P. F. Mah. From this it follows that $ {C_1}(\mathcal{H})$ has the weak $ *$-fixed point property.

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