Timeline for Does the second Bourgain–Delbaen space belong to C_p?
Current License: CC BY-SA 4.0
11 events
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| Nov 24 at 17:47 | history | edited | LSpice | CC BY-SA 4.0 |
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| Nov 24 at 16:40 | comment | added | Kevin Casto | Is it standard terminology to group these spaces into a set (well, proper class) and use $C_p$ as a noun? I have to say, almost always properties of structures (groups, rings, Banach spaces) are just left as adjectives. | |
| Nov 24 at 13:20 | answer | added | S Argyros | timeline score: 2 | |
| Nov 22 at 18:10 | comment | added | S Argyros | 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 $C_p$. | |
| Nov 22 at 13:53 | comment | added | Ioana Ghenciu | Is $\ell_2$ embedded in $Y$ by Theorem 4.2 in the paper? Then $Y\not \in C_2$ and thus $Y\not \in C_p$, $p\ge 2$. | |
| Nov 22 at 13:47 | comment | added | Ioana Ghenciu | 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. | |
| Nov 21 at 22:53 | comment | added | S Argyros | 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/… ). | |
| Nov 21 at 13:36 | comment | added | Ioana Ghenciu | 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 $Y\in C_2$? | |
| Nov 21 at 12:42 | history | edited | Ioana Ghenciu | CC BY-SA 4.0 |
added 43 characters in body
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| Nov 20 at 23:02 | comment | added | Bill Johnson | 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 | |
| Nov 20 at 13:43 | history | asked | Ioana Ghenciu | CC BY-SA 4.0 |