A subspace $Y$ of $X$ is said to be semi M-ideal if $\exists$ a projection $P$ (not necessarily linear) from $X^\*$ to $Y^\perp$ such that $\|x^\*\|=\|Px^\*\|+\|x^\*-Px^\*\|$. And also, $P(\lambda x^\*+Py^\*)=\lambda Px^\*+Py^\*$ , $\forall x^\*, y^\*\in X^\*$. 


It is known that $ker(\mathbb{1})$ as a subspace of $\ell\_1$ is a semi M-ideal, where $\mathbb{1}\in \ell\_\infty$. My question is what is the corresponding projection $P:\ell\_\infty\to sp\{\mathbb{1}\}$ which satisfies above norm decomposition?



Why ker(1) is a semi M-ideal in $\ell_1$?
For $a=(a\_n) \in \ell\_\infty$ let $m(a) = (\sup a\_n + \inf a\_n)/2$. Then $a\mapsto P(a)=m(a) 1$ is the projection you're looking for.


Dirk


