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 e > 0 we can find a ó in (0, 1) such that every weak '-compact, convex subset C of the unit ball of X whose measure of norm compactness exceeds e must meet the (1-<5)-ball of X. We show in this paper that Cx(ß7), the space of trace class operators on an arbitrary infinite-dimensional Hilbert space %? is UKK*. Consequently Cx(ßf) has weak "-normal structure. This answers affirmatively a question of A. T. Lau and P. F. Mah. From this it follows that C, (ßf) has the weak '-fixed point property.

On strong asymptotic uniform smoothness and convexity

“…Lennard proved in [20] that the trace class operators N (ℓ 2 , ℓ 2 ) has the weak* fixed point property. This result was extended by Besbes [3] to N (ℓ p , ℓ q ) with p −1 + q −1 = 1.…”

“…Lennard proved in [20] that the space of trace class operators on a Hilbert space is weak* AUC. Equivalently, K(ℓ 2 ) is AUS.…”

Fixed point property and normal structure for Banach spaces associated to locally compact groups

“…Then the space ⊕T (H f ) can be isometrically embedded into T (H). By a result of Lennard [18] T (H) has property U KK . It follows that T (H), and so every subspace of it, has the fpp.…”

Fixed point property and the Fourier algebra of a locally compact group

“…In particular, by the uniqueness of the predual of a von Neumann algebra [35, p.135], A * is isomorphic to the 1 -sum ⊕J(H f ), f ∈ ext (S), where J(H f ) denotes the space of trace-class operators on H f , which can be embedded as a subspace of J(H), where H is the Hilbert space direct sum of H f , f ∈ ext (S). By a result of Lennard [27], J(H) has UKK. In particular, every subspace of J(H) has UKK.…”