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We consider a Banach space $X$ and its dual $X^*$.

Let $Q\colon X^\ast \to X^\ast$ be an idempotent operator. Question: Can we find an idempotent operator $P\colon X^\ast \to X^\ast$ which is weak${}^\ast$ weak${}^\ast$-to-weak${}^\ast$ continuous and with range isomorphic to range of $Q$ and $\mbox{im}P\subseteq \mbox{im}Q$? In fact, I am mostly interested in the case $\mbox{im }Q\cong \ell_p$ for $p\in [1,\infty)$.

Certainly, $P$ would have to be an adjoint to some idempotent on $X$. My feeling is that in general this is not the case but perhaps it might be true for some well-behaved class of Banach spaces $X$ like Banach lattices? Or Banach lattices without a complemented copy of $\ell_1$?
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Weak*-closed and complemented subspaces of dual Banach spaces

We consider a Banach space $X$ and its dual $X^*$. Let $Q\colon X^\ast \to X^\ast$ be an idempotent operator. Can we find an idempotent operator $P\colon X^\ast \to X^\ast$ which is weak${}^\ast$ continuous and with $\mbox{im}P\subseteq \mbox{im}Q$? Certainly, $P$ would have to be an adjoint to some idempotent on $X$. My feeling is that in general this is not the case but perhaps it might be true for some well-behaved class of Banach spaces $X$ like Banach lattices? Or Banach lattices without a complemented copy of $\ell_1$?