Timeline for An abstract characterisation of weak* topologies
Current License: CC BY-SA 4.0
11 events
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| Dec 20, 2022 at 12:39 | comment | added | HardyHulley | Thanks @Mikael for the very comprehensive answer and to everyone else for their insights. I expected that the answer would be negative, but it is good to see the argument laid out clearly. I'm curious, though, about what happens with Banach spaces, such as $L^1(0,1)$, which have no predual. | |
| Dec 15, 2022 at 8:34 | history | edited | Matthew Daws | CC BY-SA 4.0 |
Fix arXiv link
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| Dec 15, 2022 at 4:47 | vote | accept | HardyHulley | ||
| Dec 14, 2022 at 23:54 | comment | added | Mikael de la Salle | I changed the phrasing to make this clear. | |
| Dec 14, 2022 at 23:52 | history | edited | Mikael de la Salle | CC BY-SA 4.0 |
Changed isomorphic to isometrically isomorphic (cd Dirk Werner's comment).
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| Dec 14, 2022 at 23:47 | comment | added | Mikael de la Salle | Thanks to both for the clarification. I know that this is clear to you, but in case somebody less familiar read these comments, let me just point out that the mere fact that $\ell_1$ has two non-isometric preduals (eg $c$ and $c_0$) is enough to say that it admits two distinct weak-* topologies.and answer the question. | |
| Dec 14, 2022 at 20:11 | comment | added | Robert Furber | Nonetheless, it is my favourite example where $\aleph_1$ occurs instead of $\mathfrak{c}$, there are $\aleph_1$ separable commutative C$^*$-algebras up to isomorphism (but $\mathfrak{c}$ up to isometry). | |
| Dec 14, 2022 at 20:04 | comment | added | Robert Furber | We can get a predual of $\ell^1$ not isomorphic to $c_0$ by trying $C(\omega^{\omega}+1)$, though I don't know a simple way of proving the non-isomorphism, only the Szlenk index. | |
| Dec 14, 2022 at 18:06 | comment | added | Dirk Werner | Actually, $c$ and $c_0$ are isomorphic, but not isometrically isomorphic. | |
| Dec 14, 2022 at 15:09 | history | edited | Mikael de la Salle | CC BY-SA 4.0 |
Typos corrected
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| Dec 14, 2022 at 8:25 | history | answered | Mikael de la Salle | CC BY-SA 4.0 |