Timeline for $C[0,1]$ fails the property (K)
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Sep 21, 2020 at 14:16 | answer | added | Dongyang Chen | timeline score: 1 | |
| Sep 21, 2020 at 13:54 | comment | added | Dongyang Chen | I give a detailed proof of the equivalence of my property (K) and the Kalton-Pelczynski version of property (K) in the following. | |
| Sep 21, 2020 at 4:11 | comment | added | Bill Johnson | Your property (K) uses the topology of uniform convergence on weakly compact sets, while the Kalton Pelczynski version of property (K) uses the topology of uniform convergence on weakly convergent sequences. How do you see that these two definitions are equivalent? Aviles and Rodriguez do not address this in their paper. | |
| Sep 21, 2020 at 0:38 | comment | added | Dongyang Chen | Yes. Kalton and Pelczynski's definition of property (K) is equivalent to my definition. | |
| Sep 21, 2020 at 0:35 | comment | added | Dongyang Chen | In your paper with Figiel and Pelczynski, a weakening of property (K), called property (k), was introduced and proved that a Banach space $X$ would fail property (k) if $X$ contains a complemented copy of $c_{0}$. This answers my question. | |
| Sep 21, 2020 at 0:30 | comment | added | Dongyang Chen | A. Aviles and J. Rodriguez (Convex combinations of weak*-convergent sequences and the Mackey topology, Mediterr J. Math. 2016) mentioned that property (K) was invented by Kwapien, but did not give the reference. | |
| Sep 20, 2020 at 17:22 | comment | added | Bill Johnson | Kalton, N. J.(1-MO); Pełczyński, A.(PL-PAN) Kernels of surjections from ℒ1-spaces with an application to Sidon sets. Math. Ann. 309 (1997), no. 1, 135--158. | |
| Sep 20, 2020 at 17:21 | comment | added | Bill Johnson | Oh, we defined a property called property (k) that is (probably) different from property (K). Kalton and Pelczynski defined property (K) somewhat differently; namely, where the convex block subsequence $y_n^*$ satisfies $y_n^*(z_n) \to 0$ for every weakly null sequence $(z_n)$ in $X$. Is that equivalent to your definition of property (K)? | |
| Sep 20, 2020 at 16:48 | comment | added | Bill Johnson | It would help if you would give references when you mention relatively obscure concepts. Where did Kwapien introduce property (K). I thought it was first defined in my paper with Figiel and Pelczynski.$$ $$ Figiel, Tadeusz; Johnson, William B.; Pełczyński, Aleksander Some approximation properties of Banach spaces and Banach lattices. Israel J. Math. 183 (2011), 199--231. | |
| Sep 20, 2020 at 2:49 | history | edited | Dongyang Chen | CC BY-SA 4.0 |
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| Sep 20, 2020 at 2:47 | history | edited | LSpice | CC BY-SA 4.0 |
Proofreading
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| Sep 20, 2020 at 1:36 | history | asked | Dongyang Chen | CC BY-SA 4.0 |