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$\DeclareMathOperator\Lip{Lip}$Let $X$ be a real Banach space. The dual $X^*$ is a closed subspace of $\Lip_0(X)$. ($\Lip_0(X)$ denotes the space of real-valued Lipschitz functions $f:X\to\mathbb{R}$ satisfying $f(0)=0$ with Lipschitz constant as norm).

Is there anything known about the complementability of $X^*$ inside $\Lip_0(X)$? If $X$ is finite dimensional, then $X^*$ is trivially complemented in $\Lip_0(X)$. Are there any examples of infinite dimensional spaces $X$ whose duals are or are not complemented in $\Lip_0(X)$?

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    $\begingroup$ When $X = L_1(\mu)$ (or more generally, $X$ is an abstract $\mathscr{L}_1$-space), then $X^*$ is injective hence complemented wherever embedded. $\endgroup$
    – Tomasz Kania
    Commented Oct 13, 2021 at 9:42

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Corollary 5.4 in my book (Lipschitz Algebras, 2nd edition): Let $V$ be a Banach space. Then there is a norm 1 linear projection from ${\rm Lip}_0(V)$ onto $V^*$. If $V$ is separable then there is a weak* continuous norm 1 linear projection.

This follows from Theorem 2 of "On nonlinear projections in Banach spaces" by Lindenstrauss (Michigan Math J 11 (1964), 263-287).

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    $\begingroup$ Thank you, Nik. I should have thought of checking the 2nd edition. I don't recall seeing this in 1st edition. $\endgroup$
    – Miek Messerschmidt
    Commented Oct 13, 2021 at 12:38
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    $\begingroup$ That's right, it's not in the 1st edition. (There's a lot more in the 2nd edition. It's twice as long!) $\endgroup$
    – Nik Weaver
    Commented Oct 13, 2021 at 12:42

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