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The second Bourgain–Delbaen space $Y$ is a separable $\mathcal{L}_\infty$ space such that $Y$ contains no copy of $c_0$ or $\ell_1$. The dual space is isomorphic to $\ell_1$.

An operator $T:X\to Y$ is $p$-convergent if it takes weakly $p$-summable sequences to norm null ones.

A Banach space $X\in C_p$ if the identity map $i$ on $X$ is $p$-convergent.

It is known that $X\in C_p$ if and only if every operator $T:\ell_{p^*}\to X$ is compact, where $p^*$ is the conjugate of $p$.

Is $Y\in C_p$ for some $1<p<\infty$ ? Is $Y\in C_2$?

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    $\begingroup$ There are many Bourgain-Delbaen spaces. For each $1<p<\infty$, there is one that is hereditarily $\ell_p$. See Haydon's paper matwbn.icm.edu.pl/ksiazki/sm/sm139/sm13935.pdf $\endgroup$
    – Bill Johnson
    Commented Nov 20 at 23:02
  • $\begingroup$ Thanks for the paper. I should have asked: if $Y$ is a separable $\mathcal{L}_\infty$ space that does not contain $c_0$ and $\ell_1$ and whose dual is isomorphic to $\ell_1$, do we know if $Y\in C_p$, for some $1<p<\infty$? In particular, can we say if $Y\in C_2$? $\endgroup$
    – Ioana Ghenciu
    Commented Nov 21 at 13:36
  • $\begingroup$ If by a B-D space we mean a $ \mathcal L_\infty $space with separable dual and the question is if such a space belongs to some $C_p$ then the answer must be negative due to the following result. Every reflexive space is embedded into a $ \mathcal L_\infty $ space with separable dual.(see sciencedirect.com/science/article/pii/… ). $\endgroup$
    – S Argyros
    Commented Nov 21 at 22:53
  • $\begingroup$ I mean $Y$ is a separable $\mathcal{L}_\infty$ space whose dual is isomorphic to $\ell_1$ constructed in the paper ``A Special class of $\mathcal{L}_\infty$ spaces, by J. Bourgain and F. Delbaen, Acta Math 145 (1980), 155-176. $\endgroup$
    – Ioana Ghenciu
    Commented Nov 22 at 13:47
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    $\begingroup$ The B-D spaces with separable dual are depended by two parameters $\alpha, \beta $ and they are $\ell_r$ saturated where the $r$ is depended by $\alpha, \beta $. It seems that the following holds. Every weakly null sequence in the $\ell_r$ saturated space has a subsequence admitting a lower $\ell_r$ estimate.For more related to this see arxiv.org/pdf/1003.0579. If this is true then indeed every such a space belongs to some $C_p$. $\endgroup$
    – S Argyros
    Commented Nov 22 at 18:10

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Since my last comment is not completely clear I will add an answer which clarifies a little more the content of the comment.

The following holds.

Proposition: Let $X_{a,b}$ be the B–D space built on the parameters $a$, $b$. Let also $r >0$ such that $\frac{1}{r} + \frac{1}{r'}=1$ and $a^{r'} +b^{r'} = 1$. Then every seminormalized weakly null sequence in $X_{a,b}$ has a subsequence that admits a lower $\ell_r$ estimate.

The proof of the proposition follows from Proposition 4.4 and Proposition 6.4 in Gasparis, Papadiamantis, and Zisimopoulou - More $\ell_r$ saturated $\mathcal{L}_\infty$ spaces, mentioned above, and a standard sliding hump argument for the weakly null sequence.

Now it follows that $X_{a,b} \in C_p$ with $p'>r$.

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