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As is known, for any compact space $T$ the Banach space $C(T)$ of all continuous functions on $T$ has the approximation property (see e.g. Albrecht Pietsch, Operator ideals). Is the same true for the (Hausdorff) quotient spaces of $C(T)$?

Does every quotient space $C(T)/X$ of $C(T)$ (where $X$ is an arbitrary closed subspace in $C(T)$, not necessarily an ideal) has the approximation property?

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  • $\begingroup$ Take a look here $\endgroup$ – Norbert Nov 23 '17 at 15:52
  • $\begingroup$ Hm... I did not understand what paper @BillJohnson speaks about. Does this mean that the answer to my question is "no"? $\endgroup$ – Sergei Akbarov Nov 23 '17 at 17:03
  • $\begingroup$ The other way round. Bill Jhonson certain that counterexample exists and may be found in A. Szankowski's paper. $\endgroup$ – Norbert Nov 23 '17 at 17:53

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