Does every contractively complemented subspace of $c\_0(I)$ is isometric to $c\_0(J)$ for some $J\subseteq I$? 


May be someone has a counterexample?



Contractively complemented subspaces of $c_0(I)$
Yes, it is true, but I don't know where it is proved in the literature. The reason I say yes is that what you want is a corollary of a unpublished almost isometric theorem that Zippin proved in the 1970s! Here is either an outline of a proof or complete nonsense: Let $P$ be a contractive projection on $c\_0(I)$. Then the range of $P^\*$ is a sublattice of $\ell\_1(I)$ (see H. Elton Lacey's book, Springer Grundlehren Band 208, The isometric theory of the classical Banach spaces) and hence is the norm closed span of a set $x\_\alpha$ of disjointly supported unit vectors. For each $\alpha$, compose $P^\*$ with the restriction mapping $R\_\alpha$ onto the support of $x\_\alpha$ to get a weak$^\*$ continuous contractive projection onto the span of $x\_\alpha$. So there is $y\_\alpha$ in $c\_0(I)$ s.t. $R\_\alpha P^\* = x\_\alpha\otimes y\_\alpha$. Argue that $(y\_\alpha)\_\alpha$ is 1-equivalent to the $c\_0(J)$ basis for some $J$ and that the range of $P$ is the closed span of $(y\_\alpha)\_\alpha$


Notice that for this to happen each $x\_\alpha$ must achieve its norm and thus has finite support. Interestingly, the $y\_\alpha$ need not have finite support.


