$L^{p}(G,X*)$ comme sous-espace complémenté de $L^{q}(G,X)*$

Mohammad Daher

Colloquium Mathematicae, Tome 131 (2013), p. 273-286 / Harvested from The Polish Digital Mathematics Library

Résumé

Let G be a compact metric infinite abelian group and let X be a Banach space. We study the following question: if the dual X* of X does not have the Radon-Nikodym property, is $L^{p}(G,X*)$ complemented in $L^{q}(G,X)*$, 1 < p ≤ ∞, 1/p + 1/q = 1, or, if p = 1, in the subspace of C(G,X)* consisting of the measures that are absolutely continuous with respect to the Haar measure? We show that the answer is negative if X is separable and does not contain ℓ¹, and if 1 ≤ p < ∞. If p = 1, this answers a question of G. Emmanuele. We show that the answer is positive if X* is a Banach lattice that does not contain a copy of c₀, 1 ≤ p < ∞. It is also positive, by a different method, if p = ∞ and X* = M(K), where K is a compact space with a perfect subset. Moreover, we examine whether $C_{Λ}(G,X*)$ may be complemented in $L_{Λ}^{∞}(G,X*)$, where Λ is a subset of Γ, the dual group of G, when the space X is separable and $L¹(G,X)/L¹_{Λ^{c}}(G,X)$ does not contain ℓ¹.

Détail

Publié le : 2013-01-01
EUDML-ID : urn:eudml:doc:286271