Quasinormal structures for certain spaces of operators on a Hilbert space

Abstract

Let E be a dual Banach space. E is said to have quasi-weak*-normal structure if for each weak* compact convex subset K of E there exists x e K such that ||* -y\\ < άiam(K) for all y e K. E is said to satisfy Lim's condition if whenever { x a } is a bounded net in E converging to 0 in the weak* topology and lim ||x α || = s then lim α \\x a + y\\ = s + \\y\\ for any y e E. Lim's condition implies (quasi) weak*-normal structure. Let H be a Hilbert space. In this paper, we prove that &~(H), the space of trace class operators on //, always has quasi-weak*-normal structure for any H; tΓ(H) satisfies Lim's condition if and only if H is finite dimensional. We also prove that the space of bounded linear operator on H has quasi-weak*-normal structure if and only if H is finite dimensional; the space of compact operators on // has quasi-weak-normal structure if and only if H is separable. Finally we prove that if X is a locally compact Hausdorff space, then C 0 (X)* satisfies Lim's condition if and only if C 0 (X)* is isometrically isomorphic to /i(Γ) for some non-empty set Γ.

Normal Structure in Dual Banach Spaces Associated With a Locally Compact Group

“…Consequently, ^(^) has weak*-normal structure. This answers affirmatively a question raised by the authors in [24]. In this section we show that there is a class of G*-algebra whose dual has property (UKK*), from which we can recover Lennard's result.…”

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

“…It follows that 1 (Γ, ω) also satisfies Lim's condition. So 1 (Γ, ω) has the RNP and weak * -normal structure (Lemma 4 in [22]). Consequently it has the weak * fpp [28].…”

“…Proof. We know that 1 (Γ) satisfies Lim's condition [28] (see also [22]), i.e., whenever f α is a net in 1 (Γ) = c 0 (Γ) * , with f α → 0 in the weak * -topology, and lim…”

Local isometries of $\mathcal {L}(X,C(K))$

“…It is worth recalling that K( 2 ) has the Namioka-Phelps property [9] and any surjective isometry of the dual is weak * -continuous.…”