Complemented subspaces of $c_0(\Gamma)$

Question

Let $\Gamma$ be an infinite set. By $c_0(\Gamma)$ we denote the Banach space of all functions $f: \Gamma \to \mathbb{R}$ such that, for all $\varepsilon > 0$, the set $\{ \gamma \in \Gamma : |f(\gamma)| \geq \varepsilon\}$ is finite, equipped with the norm $$\|f\|_{\infty} = \sup_{\gamma \in \Gamma} |f(\gamma)|.$$

In particular, $c_0 := c_0(\mathbb{N})$ is the space of real sequences that converge to zero.

It's known that every (closed) subspace of $c_0$ that is isomorphic to $c_0$ is also complemented in $c_0$. The proofs I know use Sobczyk's Theorem (http://www.ams.org/journals/proc/1971-028-02/S0002-9939-1971-0275122-0/S0002-9939-1971-0275122-0.pdf).

Is a similar result valid for uncountable $\Gamma$, namely: Let $\Gamma$ be an uncountable cardinal. Is every subspace of $c_0(\Gamma)$ that is isomorphic to $c_0(\Gamma)$ complemented in $c_0(\Gamma)$?

(note that Sobczyk's Theorem cannot be used in the uncountable (not-separable) case)

I suspect the answer is no. Any help is greatly appreciated.

Answer 1

The answer is yes. See discussion in the end of the paper

Sobczyk's theorem from A to B. F. C. Sanchez, J. M. F. Castillo, D. Yost

Answer 2

This also follows from Proposition 2.8 in

S. A. Argyros, J. F. Castillo, A. S. Granero, M. Jimenéz, and J. P. Moreno, Complementation and embeddings of $c_0(I)$ in Banach spaces, Proc. London Math. Soc. 85 (2002), no. 3, 742–768.