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

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.