Timeline for If the closed unit ball of Banach space has at least one extreme point, must the Banach space the be a dual space?
Current License: CC BY-SA 4.0
16 events
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| Dec 14, 2018 at 10:15 | comment | added | Masayoshi Kaneda | @Idonknow: The question is already answered extensively, but let me add one quick example. The identity of any unital $C^*$-algebra is an extreme point of its closed unit ball, but, of course, not all unital $C^*$-algebras are von Neumann algebras (=$C^*$-algebras with Banach space predual). | |
| Dec 4, 2018 at 11:52 | vote | accept | Idonknow | ||
| Dec 2, 2018 at 23:09 | history | edited | Idonknow | CC BY-SA 4.0 |
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| Dec 2, 2018 at 22:25 | answer | added | Bill Johnson | timeline score: 10 | |
| Dec 2, 2018 at 22:04 | comment | added | Dirk Werner | @TarasBanakh: There are infinite compact $K$ for which $C(K)$ is a dual space: these are precisely the hyperstonean $K$, e.g., $\beta\mathbb{N}$. (On the other hand there are non-dual $C(K)$ for which the unit ball is the norm-closed convex hull of its extreme points, e.g. $\alpha\mathbb{N}$. These are precisely the totally disconected $K$.) | |
| Dec 2, 2018 at 19:59 | history | edited | YCor |
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| Dec 2, 2018 at 14:54 | history | edited | Idonknow | CC BY-SA 4.0 |
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| Dec 2, 2018 at 9:33 | answer | added | MSMalekan | timeline score: 8 | |
| Dec 2, 2018 at 7:42 | comment | added | Idonknow | @MartinSleziak Yes, I mean the closed unit ball of $X.$ | |
| Dec 2, 2018 at 7:42 | history | edited | Idonknow | CC BY-SA 4.0 |
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| Dec 2, 2018 at 7:20 | comment | added | Martin Sleziak | This post on Mathematics site seems to be about the same question: Krein-Milman and dual spaces. | |
| Dec 2, 2018 at 7:13 | comment | added | Martin Sleziak | I have added the tag (extreme-points), since it seems to me a good fit to the question. There exists also (krein-milman-theorem) tag, but that one would probably be a stretch. | |
| Dec 2, 2018 at 7:11 | comment | added | Martin Sleziak | When you say that "$X$ has at least one extreme point" do you mean that the closed unit ball of $X$ has at least on extreme point? | |
| Dec 2, 2018 at 7:11 | history | edited | Idonknow | CC BY-SA 4.0 |
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| Dec 2, 2018 at 7:09 | history | edited | Martin Sleziak |
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| Dec 2, 2018 at 7:06 | history | asked | Idonknow | CC BY-SA 4.0 |