TITLE: Contractively complemented subspaces of $c_0(I)$
QUESTION [5 upvotes]: 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?

REPLY [2 votes]: 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 $c_0(J)$?