Let $X$ be a Banach space and let $p\in (1,\infty)$. If $q$ denotes the conjugate exponent to $p$, then $L\_q(X^\*)$ is easily seen to be isometric to a subspace of $(L\_p(X))^\*$ via the map $$f\mapsto \int\limits\_{[0,1]}\langle \cdot(\omega), f(\omega)\rangle\,{\rm d}\omega\quad (f\in L\_q(X^\*)).$$ One of the numerous characterisations of [the Radon–Nikodym property](https://en.wikipedia.org/wiki/Bochner_integral#Radon.E2.80.93Nikodym_property) asserts that $X^\*$ has this property with respect to $[0,1]$ endowed with the Lebesgue measure if and only if this map is surjective.



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> Suppose that $X^\*$ fails the Radon–Nikodym property with respect to the Lebesgue measure. Is the range of the above-mentioned map complemented? 
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My motivation follows from [this question of M. González](https://mathoverflow.net/questions/225081/banach-spaces-x-with-ell-2x-not-isomorphic-to-l-20-1-x/225757#225757); if the answer were positive, then we would have $\ell\_2(X)\not\cong L\_2(X)$ for every infinite-dimensional space $X$ with $X^\*\cong L\_1(\mu)$ and for every infinite-dimensional C\*-algebra.


Edit (26.01.2016): This is indeed a known result. It follows from Theorem 9 in:



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> Z. Hu and B.-L. Lin, [Extremal structure of the unit ball of $L\_p(\mu, X)^\*$](http://www.sciencedirect.com/science/article/pii/S0022247X96902255), *J. Math. Anal. Appl.* **200** (1996), 567–590.
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Is $L_q(X^*)$ complemented in $(L_p(X))^*$?
Still no answers? Perhaps we should try the following. 


Let $g : [0,1] \to X^\*$ be weak\* measurable, in the sense that, for every $x \in X$, the function
$$
\omega \mapsto \langle x, g(\omega)\rangle
$$
is measurable. Then there exist weak\* measurable functions $g\_1, g\_2 : [0,1] \to X^\*$ such that 


(1) $g$ is weak\* equivalent to $g\_1+g\_2$, in the sense that, for every $x \in X$,
$$
\langle x,g(\omega)\rangle = \langle x,g\_1(\omega)+g\_2(\omega)\rangle
$$
for almost all $\omega \in [0,1]$. (But the exceptional nullset depends on $x$.) 


(2) $g\_1$ is Bochner measurable in the sense that the range of $g\_1$ is separable. 


(3) $g\_2$ is totally non-separable in the sense that: for every separable $Y \subseteq X^\*$, the outer measure of $g\_2^{-1}(Y)$ is zero. 


If this works, then perhaps the elements of
$(L\_p(X))^\*$ can be identified as such weak\* measurable functions $g$, and the projection is the map sending $g$ to its "Bochner part" $g\_1$.


