$L^p (G, X^*)$ comme sous-espace complémenté de $L^{q}(G, X)^{*}$
Volume 131 / 2013
Abstract
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 $\ell ^1$.
