Timeline for What is the Banach-Mazur distance between $\ell_\infty$ and $L_\infty$?
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21 events
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| Sep 13, 2021 at 22:21 | comment | added | James E Hanson | @YemonChoi That is basically what I am wondering, although I wouldn't necessarily phrase it in terms of axioms. I would phrase it like this: Is there a rational number $r$ such that $d(\ell_\infty,L_\infty) < r$ and $d(\ell_\infty,L_\infty) > r$ are both consistent with ZFC? | |
| Sep 13, 2021 at 1:56 | comment | added | Yemon Choi | @JamesHanson I'm not sure I follow (since I am a caveman when it comes to set theory). Is your question whether additional axioms might allow one to "construct" isomorphisms $\ell_\infty \leftrightarrow L_\infty$ that have smaller norm than ones constructed with "bare" ZFC? | |
| Sep 10, 2021 at 21:15 | comment | added | James E Hanson | Is there an argument that ZFC actually uniquely specifies the value of $d(\ell_\infty,L_\infty)$? | |
| Sep 10, 2021 at 13:09 | comment | added | YCor | Does the proof provide an upper bound? | |
| Sep 10, 2021 at 13:08 | history | edited | YCor |
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| Sep 10, 2021 at 12:54 | answer | added | Fred Dashiell | timeline score: 2 | |
| Jun 14, 2016 at 9:38 | comment | added | Hannes Thiel | @YemonChoi That is a nice idea. Certainly $X=L_\infty$ and $Y=\ell_\infty$ are injective Banach lattices. It looks like the embedding $\ell_\infty \to L_\infty$ (from the post of Theo Buehler) does identify $\ell_\infty$ with a sublattice of $L_\infty$. Thus, one would get a positive projection from $X$ onto $Y$. For the embedding $L_\infty \to \ell_\infty$ this is not clear to me. Maybe it is not possible to embed $L_\infty$ as a closed sublattice of $\ell_\infty$. There are structure results for lattice homomorphism $C(K_1)\to C(K_2)$, see e.g. 3.2.10 in "Banach Lattices" by Meyer-Nieberg. | |
| Jun 14, 2016 at 8:28 | comment | added | Yemon Choi | @HannesThiel I admit I did not think this through very carefully. My idea (which may have a silly mistake) was to get a positive norm-1 projection $P$ from $L_\infty$ onto $\ell_\infty$, in which case $I-P$ should also be a positive norm-1 projection (I think?) onto the complementary subspace. | |
| Jun 13, 2016 at 20:26 | comment | added | Hannes Thiel | @YemonChoi Following the construction from the post of Theo Buehler, I understand that $d(X,X\oplus_\infty Y)\leq 4$, and similarly $d(X\oplus_\infty Y,Y)\leq 4$, for $X=L_\infty$ and $Y=\ell_\infty$. But this only gives $d(X,Y)\leq 16$, Do you think that $d(X,X\oplus_\infty Y)\leq 2$? Since $X$ and $X\oplus_\infty Y$ are not isometrically isomorphic, we have $d(X,X\oplus_\infty Y)\geq 2$. With an upper bound of $2$, we would thus get $d(X,X\oplus_\infty Y)=2$. | |
| Jun 10, 2016 at 12:43 | comment | added | Hannes Thiel | I was not aware of the Amir-Cambern theorem, which gives a nice lower bound $d(\ell_\infty,L_\infty)\geq 2$. I looked a little bit at this topic now, and found the paper "A second-dual method for C(X) isomorphisms" by Cohen, J. Funct. Anal. 23 (1976). It contains results about C(K) spaces with BM-distance $<3$. Does anybody see if this can be used to show $d(\ell_\infty,L_\infty)\geq 3$ ? | |
| Jun 9, 2016 at 15:23 | comment | added | Yemon Choi | Reading the "construction" given by Theo Buehler (see @RobertIsrael's comment) my guess would be that one can get a BM-distance of $\leq 4$ by careful book-keeping of the isomorphisms, since both $Y=\ell_\infty$ and $X=L_\infty$ are $1$-injective, and since $X\cong_1 X\oplus_\infty X$ and $Y\cong_1 Y\oplus_\infty Y$ | |
| Jun 9, 2016 at 6:24 | comment | added | Hannes Thiel | If this is relevant, I am interested in the situation assuming ZFC. | |
| Jun 9, 2016 at 6:16 | history | edited | Fedor Petrov | CC BY-SA 3.0 |
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| Jun 9, 2016 at 0:54 | comment | added | Robert Israel | See this discussion of that non-constructivity (in particular t.b.'s answer which shows that in $ZF + DC + PM_\omega$ they are not isomorphic). | |
| Jun 9, 2016 at 0:46 | comment | added | Gerald Edgar | Banach spaces $l^\infty$ and $L^\infty$ are isomorphic, but isomorphisms are not constructive. Something like the Hahn-Banach theorem is required to prove the existence of an isomorphism. | |
| Jun 8, 2016 at 23:52 | answer | added | Bunyamin Sari | timeline score: 10 | |
| Jun 8, 2016 at 21:16 | comment | added | Hannes Thiel | Yes, they are isomorphic and therefore the distance is finite. | |
| Jun 8, 2016 at 21:12 | comment | added | Bunyamin Sari | Yes, it is well known that they are isomorphic. This follows from the injectivity of these spaces and Pelczynski decomposition method. The details can be found, for instance, in Albiac-Kalton. I am not sure if the answer to the question is known though. | |
| Jun 8, 2016 at 21:12 | comment | added | Nik Weaver | @DenisSerre: $l^\infty$ and $L^\infty$ are isomorphic as Banach spaces. I think this is due to Pelczynski. | |
| Jun 8, 2016 at 21:00 | comment | added | Denis Serre | Is there any reason to think that it is finite ? | |
| Jun 8, 2016 at 19:53 | history | asked | Hannes Thiel | CC BY-SA 3.0 |