Direct proof of injectivity of $L_\infty$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T13:22:06Z http://mathoverflow.net/feeds/question/110461 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/110461/direct-proof-of-injectivity-of-l-infty Direct proof of injectivity of $L_\infty$ Norbert 2012-10-23T19:36:51Z 2013-02-12T15:14:52Z <p>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. </p> <ul> <li>$L_\infty$ as commutative unital $C^*$ algebra is isometrically isomorphic to $C(K)$ for some compact $K$.</li> <li>Every $C(K)$ space which is a dual space is isometrically injective.</li> </ul> <p>However the proof for $\ell_\infty$ is quite simple. Let $i:X\to Z$ be isometric embedding and $T:X\to Y$ be a bounded operator. Let $e_n:\ell_\infty\to\mathbb{C}:x\mapsto x(n)$ be coordiante functionals, then consider bounded functionals $f_n:\mathrm{Im}(i)\to \ell_\infty:z\mapsto e_n(T(i^{-1}(z)))$ extend them by Hahn-Banach theorem to get functionals $g_n:Z\to\mathbb{C}$. The desired operator is $\hat{T}:Z\to\ell_\infty: z\mapsto(g_1(z), g_2(z),\ldots)$</p> <p><strong>My question:</strong></p> <p>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 $(e_n:n\in\mathbb{N})\subset\ell_\infty^*$. </p> <p>Thank you.</p> http://mathoverflow.net/questions/110461/direct-proof-of-injectivity-of-l-infty/110475#110475 Answer by Bill Johnson for Direct proof of injectivity of $L_\infty$ Bill Johnson 2012-10-23T21:00:17Z 2012-10-23T21:00:17Z <p>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.  Basically the same argument works for any dual space that is $\mathcal{L}_{\infty,\lambda}$ for all $\lambda > 1$.</p> http://mathoverflow.net/questions/110461/direct-proof-of-injectivity-of-l-infty/121613#121613 Answer by jbc for Direct proof of injectivity of $L_\infty$ jbc 2013-02-12T15:14:52Z 2013-02-12T15:14:52Z <p>I am not sure whether the following would meet your requirements but I vaguely remember having heard it in a course many years ago and I think that it is sufficiently distinct from the above proofs to justify a brief mention. The crucial common property of the three spaces in question---the real line, the sequence space and the function space---is that they are Dedekind-complete Banach lattices for which the norm is intimately connected to the order structure---the unit ball coincides with the interval $[-1,1]$. This fact allows one to mimic directly the standard proof of the classical Hahn-Banach theorem in the two more advanced cases.</p>