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The question, already phrased in the title, looks like a classical problem from Banach space theory from the 1970s. Hence, my question is more of a reference request in its nature.

Can every separable Banach space with the metric approximation property be isometrically embedded as a 1-complemented subspace of a space with a monotone basis?

Perhaps this could be already achieved in some renorming of Pełczyński's universal space.

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  • $\begingroup$ I don't know, but do you know about Dor's negative result in the category of finite dimensional normed spaces? $\endgroup$
    – Bill Johnson
    Commented Dec 6, 2019 at 1:51
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    $\begingroup$ Dor, L. E. A note on monotone bases. Israel J. Math. 14 (1973), 285–286. $\endgroup$
    – Bill Johnson
    Commented Dec 6, 2019 at 1:51

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