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Bourgain/Casazza/Lindenstrauss/Tzafriri proved in their unconditional basis UTAP book (1985) that $\ell_1$ is the only nontrivial complemented subspace of $(\bigoplus\ell_2^n)_1$, and hence by duality $c_0$ is the only nontrivial complemented subspace of $(\bigoplus\ell_2^n)_0$. And of course there are no nontrivial complemented subspaces of $(\bigoplus\ell_2^n)_p$ for $1<p<\infty$ due to the fact that it's just $\ell_p$ itself.

I am wondering about the general case where $p\neq q\neq 2$. Is $\ell_p$ then the only nontrivial complemented subspace of $(\bigoplus\ell_q^n)_p$?

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    $\begingroup$ I guess you mean $p \not= 1$. I think your question is still open. $\endgroup$
    – Bill Johnson
    Commented Sep 3, 2020 at 19:49
  • $\begingroup$ @BillJohnson Ah okay, that's unexpected. I thought for sure this would have been solved. But are you also saying that the question is closed for $(\oplus\ell_q^n)_1$ for $q$ other than just 2? $\endgroup$
    – Ben W
    Commented Sep 3, 2020 at 20:37
  • $\begingroup$ I just meant that when $p=1$ and $1<q \not=2 < \infty$ the space contains a complemented subspace isomorphic to the $1$ sum of $\ell_2^n$, which is not isomorphic either to $\ell_1$ or to the whole space. $\endgroup$
    – Bill Johnson
    Commented Sep 3, 2020 at 21:47
  • $\begingroup$ @BillJohnson Oh, yes, the complemented version of Dvoretsky. $\endgroup$
    – Ben W
    Commented Sep 3, 2020 at 21:51

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