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Let $X$ be a set of cardinality $\aleph_1$ and consider the Banach space $\ell_\infty^c(X)$ of all scalar-valued bounded functions on $X$ which are non-zero only for countably many elements of $X$ endowed with the sup-norm. It is a nice example of a separably injective Banach space. I have two natural question concerning complemented subspaces of that space

  • Q1. Is every subspace $Y\subset \ell_\infty^c(X)$ isomorphic to $\ell_\infty^c(X)$ complemented?
  • Q2. If not, does every subspace of $\ell_\infty^c(X)$ isomorphic to $\ell_\infty^c(X)$ contain a subspace still isomorphic to $\ell_\infty^c(X)$ which is complemented?

I was hoping to employ separable injectivity but somehow it does not work. I am almost sure that Q2 holds true.

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