On the Dieudonné property for 𝐶(Ω,𝐸)

Abstract

ABSTRACT.In a recent paper, F. Bombai and P. Cembranos showed that if E is a Banach space such that E* is separable, then C(iï,E), the Banach space of continuous functions from a compact Hausdorff space fi to E, has the Dieudonné property. They asked whether or not the result is still true if one only assumes that E does not contain a copy of fi. In this paper we give a positive answer to their question.As a corollary we show that if E is a subspace of an order continuous Banach lattice, then E has the Dieudonné property if and only if C(fi, E) has the same property.

Remarks on Weak Compactness of Operators Defined on Certain Injective Tensor Products

“…In the case of X = C{K) there are some papers devoted to the study of this question (see [2,[6][7][8][9]), but nothing seems to be known in the present setting; we observe that the theorems proved in the paper extend all of the above-quoted results, but their proofs make use of the results in [2,8], so that they may be considered interesting complements to those theorems. Because the proofs of our results are similar, we give the proof of Theorem 2 only and leave the others to the reader.…”

“…Because the proofs of our results are similar, we give the proof of Theorem 2 only and leave the others to the reader. We need the following definition: a Banach space E has the Dieudonné property if any weakly completely continuous (or Dieudonné) operator defined on it is weakly compact [8].…”

“…The density of the elements of the type £f_, x** ®y, in A** ®E Y and the continuity of T on A** <gi£ Y give our claim: T takes its values in Z . Now recall that A** has the metric approximation property (see [3]) and so A** ®£ Y = Kw.iY*, A**) (= the Banach space of all w* -w continuous compact operators from Y* into A**) ; furthermore, A** is complemented in some C{K) space [1] and so KW-{Y*, A**) is complemented in C{K, Y) by a projection P. The operator T o P: CiK, Y) -> Z is a Dieudonné operator that must be weakly compact because of the result of [8]. Since To p restricted to X ®eY is nothing else than T, we are done.…”

Some stability properties of Arens regular bilinear operators

“…As an application of this result, we give a very short proof of the known fact that, for an algebra A not containing an isomorphic copy of I 1 , the algebra C(K, A) is Arens regular iff A is so [24]. The main ingredients of the proofs are a result of J. Diestel [7] about weak compactness in the space L l {n, X) of Bochner integrable functions, some results of A. Grothendieck [12] and N. J. Kalton, E. Saab and P. Saab [15] about the Dieudonne property, and Choquet's integral representations Theorem [19].…”

Another proof of a result of N. J. Kalton, E. Saab and P. Saab on the Dieudonné property in C(K, E)

“…Let K be a compact Hausdorff topological space and £ be a Banach space not containing Z 1 . Recently N. J. Kalton, E. Saab and P. Saab ( [5]) obtained the results that under the above assumptions the usual space C{K, E) has the Dieudonn6 property; i.e. each weakly completely continuous operator on C(K, E) is weakly compact.…”