An alternative Dunford-Pettis Property

The Alternative Dunford-Pettis Property in Subspaces of Operator Ideals

“…A similar result, among other things, will be found about Banach spaces with the DP or DP1 property in [10,Theorem 1] and [12,Theorem 1.4]. Specially, a Banach space X has the DP property if and only if every weakly compact operator T : X → Y is completely continuous; while X has the DP1 property if and only if every weakly compact operator T : X → Y is DP1.…”

“…Recall that, by Freedman's Theorem [12], an arbitrary Banach space X has the DP1 property if and only if every weakly compact operator T from X into arbitrary Banach space Y is DP1. So, in order to prove a key result of this article, one can give a necessary and sufficient condition among Banach spaces containing no copy of l 1 to have the DP1.…”

The Alternative Dunford–Pettis Property for Subspaces of the Compact Operators

“…Indeed, let {e n } be an orthonormal sequence in a Hilbert space H. The closed subspace M of K(H) generated by {e * 1 ⊗ e 1 , e * n ⊗ e 2 : n ≥ 2} is isometric to R ⊕ ∞ 2 and e * n ⊗ e m (h) := (h|e n )e m . Therefore M does not satisfy the DP 1 property (see Freedman [12, Example 1.6]), however M * satisfies the KKP since it is isometric to R ⊕ 1 2 (see Freedman [12,Theorem 1.9]). Another example is M = K(H).…”

“…Since its introduction by Grothendieck, the DP has had an important development. We refer to Diestel [10] as an excellent survey Freedman [12] characterized Banach spaces having the DP1 as those Banach spaces X for which for any Banach space Y any weakly compact operator T : X −→ Y is a DP1 operator. Here we prove that if X is a Banach space with a shrinking Schauder basis, then for subspaces M of K(X) satisfying certain isometric assumption, it is enough to assume that the evaluation operators on M are DP1 operators in order to check that M has the DP1.…”

The alternative Dunford–Pettis Property in the predual of a von Neumann algebra

“…Let A be a type II von Neumann algebra with predual A * . We prove that A * does not have the alternative Dunford-Pettis property introduced by W. Freedman [7], i.e., there is a sequence (ϕ n ) converging weakly to ϕ in A * with ϕ n = ϕ = 1 for all n ∈ N and a weakly null sequence (x n ) in A such that ϕ n (x n ) 0. This answers a question posed in [7].…”