I am looking for a resource or some hints on why the two normed spaces $\ell_\infty$ and $L_\infty$ belong to the family $\mathcal{B}_1$, that is,
they are of the family of Banach spaces $X$ such that for any Banach space $X'$ containing $X$ there is a projection $P$ from $X'$ to $X$ such that $\lVert P\rVert\leq 1$.
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2$\begingroup$ See also math.stackexchange.com/questions/470869/… $\endgroup$– Gerald EdgarCommented Mar 29, 2021 at 20:48
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2$\begingroup$ mathoverflow.net/questions/110461/… $\endgroup$– Tomasz KaniaCommented Mar 29, 2021 at 21:10
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1 Answer
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I think the general fact is that $\mathcal{B}_1$ consists of precisely the Banach spaces (isometrically isomorphic to) the spaces $C(X)$ for $X$ extremally disconnected and compact. In particular, every $L^\infty$ space will belong to $\mathcal{B}_1$. See Projection constants and spaces of continuous functions by Isbell and Semadeni for references.
answered Mar 29, 2021 at 20:50