Timeline for Examples of non $w^{*}$-closed complemented subspaces of a dual Banach space that are also dual spaces
Current License: CC BY-SA 4.0
10 events
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| Jun 21, 2023 at 23:40 | comment | added | Bill Johnson | Perhaps interesting for you is an old observation of Joram Lindenstrauss: If $X$ is complemented in some dual space, then it is complemented in $X^{**}$. | |
| Jun 12, 2023 at 11:07 | comment | added | Onur Oktay | We may use Goldstine's theorem as general way of producing other examples. Let $Z$ be any nonreflexive Banach space, $Y=Z^*$, $X=Z^{***}$. It is well-known that $Z^*$ is a closed, complemented, weak$^*$ dense subspace of $Z^{***}$. | |
| Jun 11, 2023 at 16:55 | comment | added | Bunyamin Sari | The James space $J$ is complemented in the double dual $J^{**}=J\oplus [1]$ where $[1]=(1,1,1,, \ldots)$. $J$ is a dual space and has explicit description. There is a rich literature on the James space and its variations. Here is a book cambridge.org/core/books/james-forest/… | |
| Jun 11, 2023 at 16:34 | comment | added | Onur Oktay | I'm not an MO editor but I believe editing/closing the question is your call. I'm glad if you found this example useful. It isn't exactly a home run - perhaps as useful as pepper. As an attempt for a pitch, let $A$ be a $C^*$-algebra that contains a complemented $\ell^2$ subspace and $M=A^{**}$. This is the case whenever $A^{**}$ is not type I finite. Then $M$ also contains a complemented $\ell^2$ subspace. I'm not sure at the moment if this subspace is always weak$^*$ closed. | |
| Jun 11, 2023 at 13:58 | comment | added | Onur Oktay | I'm writing down the first example that comes to my mind, so apologies in advance for probable mistakes: Let $D\subset [0,1]$ be the set of dyadic numbers, $\delta_D = \{\delta_r: r\in D\}\subset M([0,1])$, $Y$ be the closed linear span of $\delta_D$. It's not difficult to verify that $Y$ is isomorphic to $\ell^1$, $Y$ is complemented in $M([0,1])$, and $Y$ is not weak$^*$ closed. | |
| Jun 11, 2023 at 12:43 | history | edited | Jon Bannon | CC BY-SA 4.0 |
edited tags
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| Jun 11, 2023 at 12:38 | history | edited | Jon Bannon |
edited tags
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| Jun 11, 2023 at 12:26 | history | edited | Jon Bannon | CC BY-SA 4.0 |
Added some motivation, and refined the title
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| Jun 11, 2023 at 1:37 | history | edited | YCor | CC BY-SA 4.0 |
removed capitals from title, added tag
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| Jun 11, 2023 at 1:09 | history | asked | Jon Bannon | CC BY-SA 4.0 |