I would like to know a simple proof of isometric injectivity of $L\_\infty$. The proof I've found in Topics in Banach space theory. F. Albiac, N. Kalton uses two deep result. 


* $L\_\infty$ as commutative unital $C^\*$ algebra is isometrically isomorphic to $C(K)$ for some compact $K$.
* Every $C(K)$ space which is a dual space is isometrically injective.


However the proof for $\ell\_\infty$ is quite simple. Let $i:X\to Z$ be isometric embedding and $T:X\to \ell\_\infty$ be a bounded operator. Let $e\_n:\ell\_\infty\to\mathbb{C}:x\mapsto x(n)$ be coordiante functionals, then consider bounded functionals $f\_n:\mathrm{Im}(i)\to \mathbb{C}:z\mapsto e\_n(T(i^{-1}(z)))$ extend them by Hahn-Banach theorem to get functionals $g\_n:Z\to\mathbb{C}$. The desired operator is $
\hat{T}:Z\to\ell\_\infty: z\mapsto(g\_1(z), g\_2(z),\ldots)$


**My question:**


Does there exist a direct proof that $L\_\infty$ is isometrically injective, a proof similar to the arguments used for the $\ell\_\infty$ space? The problem in mimicking proof for $\ell\_\infty$ arose from the fact that I can't find family of functionals $(E\_n:n\in\mathbb{N})\subset L\_\infty^\*$ similar to coordinate functionals $(e\_n:n\in\mathbb{N})\subset\ell\_\infty^\*$. 


Thank you.



Direct proof of injectivity of $L_\infty$
Write $L\_\infty$ as the closure of a net (directed by inclusion) of finite dimensional $\ell\_\infty$ spaces. Compose the operator into $L\_\infty$ with norm one projections onto these subspaces and extend. Use weak$^\*$ compactness of the unit ball of $L\_\infty$ to pass to a limit of a subnet of these operators.
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Basically the same argument works for any dual space that is $\mathcal{L}\_{\infty,\lambda}$ for all $\lambda > 1$.


