Contractively complemented subspaces of $c_0(I)$

Question

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?

Answer 1

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 The reason I say yes is that what you want is a corollary of a unpublished almost isometric theorem that Zippin proved in the 1970s!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.

Answer 2

As you may know already, every contractively complemented subspace of $\ell_1(I)$ is isometric to $\ell_1(J)$ for some $J \subset I$ --- see for example Semadeni, Banach Spaces of Continuous Functions, 1971, p. 488. This is just the dual of your desired proposition. If $X$ is a contractively complemented subspace of $c_0(I)$ by a projection $P$, then the adjoint $P^*$ (as in Bill's answer above) is an isometric embedding of $X^*$ into $\ell_1(I)$ such that $P^*(X)$ is contractively complemented by the restriction operator, so that $P^*(X)$ is isometric to some $\ell_1(J)$. Can you then argue by w*-continuity of the isometry that $X$ must be isometric to $c_0(J)$?