Timeline for Decomposable Banach Spaces
Current License: CC BY-SA 3.0
10 events
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| Aug 9 at 16:45 | comment | added | tj_ | BTW: Your observation, that the dual of a non-separable Banach space is decomposable, is contained in the paper of Sims, Yost: "Linear Hahn-Banach Extension Operators" from 1989 (Corollary, p. 55) with a proof along the lines you sketched in mathoverflow.net/questions/74171/… | |
| Mar 8, 2016 at 14:28 | vote | accept | Bill Johnson | ||
| Mar 8, 2016 at 13:42 | answer | added | Tomasz Kania | timeline score: 13 | |
| Aug 9, 2011 at 10:55 | comment | added | Kevin Beanland | Philip: I think the issue was that this paper is not on the arxiv. I saw him at a conference this past February and he sent me a copy, otherwise I would have missed it. I think the question is very interesting. I just did a mathsci review for the paper of Dodos, Lopez-Abad and Todorcevic I mentioned above; it's definitely worth a read, it is well-written, pretty short and gave me better understanding about how to approach these type of problems (which I had no idea about before hand). | |
| Aug 9, 2011 at 7:21 | comment | added | Philip Brooker | Kevin, thanks for mentioning Koszmider's survey paper (which I didn't think to look in for this question). I was sure I'd seen the question published somewhere before and was just about to go looking for it when your comment appeared. | |
| Aug 9, 2011 at 0:35 | comment | added | Kevin Beanland | One more related result to point out (for anyone interested) is that for HI spaces the bound is $2^\omega$; since every HI space embedds into $\ell_\infty$ (it seems that many authors independently proved this. The book of Argyros and Todorcevic contains the proof, which is not all that hard.) | |
| Aug 9, 2011 at 0:32 | comment | added | Kevin Beanland | In the paper by Piotr Koszmider; A survey on Banach spaces C(K) with few operators; RACSAM 104 (2), 2010, pp. 309 -326. He mentions this question (Problem 6) and attributes it Argyros; it sounds like the question was communicated to him personally and had not appeared in the literature before this survey. I have a copy of the paper if you want me to send it. You may want to ask Dodos, Lopez-Abad or Todorcevic as well. They have recent work (which appeared in the Advances) on finding a similar bound for spaces containing unconditional basic sequences. | |
| Aug 8, 2011 at 8:13 | comment | added | Bill Johnson |
Thanks, Philip. Somehow I missed this recent paper of Koszmider even though I knew his work on indecomposable $C(K)$ spaces. As far as I know, the question does not appear in the published literature.
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| Aug 8, 2011 at 4:52 | comment | added | Philip Brooker | Great question, but I really have no idea about the answer. Last I heard (though I haven't really read the paper, namely arxiv.org/abs/1106.2916 ) was Koszmider's result that it is consistent that there is a space $C(K)$ of density $2^c$, where $c$ is the cardinality of the continuum. In introduction to the preprint, Koszmider mentions that there is a bound on the density of the spaces having the properties that his $C(K)$ has, but that the question posed above seems to still be open. | |
| Aug 8, 2011 at 0:04 | history | asked | Bill Johnson | CC BY-SA 3.0 |