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

### Tom 131 / 2013

Colloquium Mathematicum 131 (2013), 273-286 MSC: 43A15, 46G10; Secondary 46B22. DOI: 10.4064/cm131-2-9

#### Streszczenie

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 $L^q (G, X)^*$, $1 < p \leq \infty$, $1/p + 1/q = 1$, 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 $\ell ^1$, and if $1 \leq p < \infty$. 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 $c_0$, $1 \leq p < \infty$. It is also positive, by a different method, if $p = \infty$ and $X^* = M(K)$, where $K$ is a compact space with a perfect subset.

Moreover, we examine whether $C_\varLambda (G, X^*)$ may be complemented in $L_\varLambda ^\infty (G, X^*)$, where $\varLambda$ is a subset of $\varGamma$, the dual group of $G$, when the space $X$ is separable and $L^1 (G, X) / L_{\varLambda ^c}^1 (G, X)$ does not contain $\ell ^1$.

#### Autorzy

• Mohammad Daher32 rue Jacques Monod
77350 Le Mée-sur-Seine, France
e-mail

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