Timeline for Norms of commutators
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Aug 14, 2017 at 21:38 | comment | added | Suvrit | Thanks for the update @mohanravi --- I will update my answer to reflect this changed state of affairs! | |
| Aug 14, 2017 at 21:21 | comment | added | mohanravi | We have realized that that assertion in our paper ' Asymptotically Optimal Multi-Paving' that the existence of $(O(\epsilon^{-2}),\epsilon)$ pavings for zero diagonal matrices implies a polylogarithmic estimate for the quantitative commutator problem is incorrect. We have only been able to very slightly improve the Johnson-Ozawa-Schechtman result: We are able to show that given any zero diagonal $A \in M_m(\mathbb{C})$, there is a representation $A = [B,C]$ such that $||B|| ||C|| \leq C \operatorname{exp}(D\sqrt{\operatorname{log}(m)}) ||A||$, where $C, D$ are universal constants. | |
| Aug 11, 2017 at 19:53 | comment | added | Suvrit | @YCor -- thanks for helping make the answer more precise!! | |
| Aug 11, 2017 at 19:53 | history | edited | Suvrit | CC BY-SA 3.0 |
finally fixed! thanks Ycor
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| Aug 11, 2017 at 19:39 | comment | added | YCor | Anyway for $A\neq 0$ apply the result to $A/\|A\|$. It yields $B',C$ with $\|B'\|,\|C\|\le K\log(n)^2$ with $[B',C]=(1/\|A\|)A$. Then, for $B=\|A\|B'$, we have $[B,C]=A$, $\|B\|\le K\log(n)^2\|A\|$, $\|C\|\le K\log(n)^2$. Hence $\|B\|\cdot\|C\|\le K^2\log(n)^4\|A\|$ and of course the case $A=0$ works too. | |
| Aug 11, 2017 at 19:35 | history | edited | Suvrit | CC BY-SA 3.0 |
added a correction and explanation
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| Aug 11, 2017 at 19:31 | comment | added | Suvrit | @YCor -- indeed, you are right -- the paper that I cite, misquotes BOS, who as you say upper bound the product not the max. I am going to update my answer to reflect this. | |
| Aug 11, 2017 at 16:02 | comment | added | YCor | In the linked paper the BOS paper looks misquoted: the upper bound is on the product $\|B\|\cdot\|C\|$, not on $\max(\|B\|,\|C\|)$ (written $\|B\|,\|C\|\le\dots$ in your link and also in your answer). Or I miss something, could you clarify? | |
| Aug 11, 2017 at 15:14 | comment | added | Suvrit | I believe existence of such an improvement was essentially prognosticated in the paper of Johnson, Ozawa, and Schechtman already. | |
| Aug 11, 2017 at 15:13 | history | answered | Suvrit | CC BY-SA 3.0 |