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A Banach space $X$ is called automorphic if for every closed subspace $Y\subseteq X$ with $\dim X/Y=\infty$, every automorphism (= linear continuous isomorphism) of $Y$ can be extended to an automorphism of $X$.

The Banach space in the following problem is assumed to be separable and infinite-dimensional.

Problem. Is every automorphic Banach space isomorphic to $c_0$ or $\ell_2$?

This problem was posed on 23 November 2022 by Anatolij Plichko on page 39 of Volume 3 of Lviv Scottish Book.

Prize. Bottle of ''Тернівка''.

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  • $\begingroup$ Do you mean "continuous linear isomorphism" instead of "linear isomorphism"? $\endgroup$
    – user473423
    Commented Dec 21, 2022 at 14:41
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    $\begingroup$ Technically the answer is no, since a non-separable Hilbert space is a counterexample (and also finite-dimensional ones...). The conclusion should probably rewritten as "isomorphic to $c_0(X)$ or $\ell^2(X)$ for some set $X$", or it should be assumed that $X$ is separable and infinite-dimensional. $\endgroup$
    – YCor
    Commented Dec 21, 2022 at 15:57
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    $\begingroup$ @YCor I think it is better assume that $X$ is separable and infinite-dimensional. Thanks. $\endgroup$
    – Lviv Scottish Book
    Commented Dec 21, 2022 at 19:00
  • $\begingroup$ I don't understand the Scottish book reference. This problem is due to Lindenstrauss and Rosenthal (link.springer.com/content/pdf/10.1007/…). It is also explicitly stated in Lindentrauss-Tzafriri's book (Problem 2.f.11). $\endgroup$
    – Bunyamin Sari
    Commented Dec 22, 2022 at 15:16
  • $\begingroup$ @BunyaminSari Indeed, this problem is very close to (but nonetheless different than) Problem 2.f.11 from Lindenstrauss-Tsafriri's textbook, which asks the following: Assume that $X$ is a separable infinite-dimensional Banach space such that for any $Y,Z$ are two isomorphic subspaces of infinite codimension in $X$ there exists an automorphism $\tau$ of $X$ such that $\tau Y=Z$. Is $X$ isomorphic to $\ell_2$ or $c_0$? $\endgroup$
    – Lviv Scottish Book
    Commented Dec 23, 2022 at 11:51

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