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## Colloquium Mathematicum

2013 | 131 | 2 | 273-286
Tytuł artykułu

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

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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.
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273-286
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2013
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• 32 rue Jacques Monod, 77350 Le Mée-sur-Seine, France
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