Timeline for Minimal injective extension is rigid
Current License: CC BY-SA 4.0
18 events
| when toggle format | what | by | license | comment | |
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| Feb 20, 2023 at 15:36 | comment | added | J. De Ro | Paulsen's book contains a proof, so I'm not sure what exactly you are looking for. See corollary 15.7. | |
| Nov 24, 2021 at 23:12 | comment | added | Andromeda | @DarthVader If you manage to fix the argument and show that $\operatorname{Fix}(\varphi)$ is injective, I would be glad to see that! | |
| Nov 24, 2021 at 22:58 | comment | added | Darth Vader | @JessePeterson: You are right in your objection. I was following Skalski's argument (isibang.ac.in/~jay/OTOA/OTOA16/slides/Adam.pdf) but he needs $\phi$ to be normal. And of course, $W$ need not be weakly closed here, so extending the map $\phi$ to $B(H)$ also doesn't help with this argument of construction of $E$. | |
| Nov 24, 2021 at 22:30 | answer | added | Jesse Peterson | timeline score: 3 | |
| Nov 24, 2021 at 21:40 | comment | added | Andromeda | @DarthVader It is not clear to me why the map you obtain from taking this limit works. Moreover, there seems to be an additional technicality. I think to apply the compactness of the space we must view the net of UCP maps in $UCP(W, B(H))$ with enlargened codomain. Then your limit is a map $W \to B(H)$. Maybe we can compose with a conditional expectation $B(H) \to W$ (exists by injectivity of $W$) to solve the latter problem, but even then it is not clear to me why it works. | |
| Nov 24, 2021 at 20:55 | comment | added | Jesse Peterson | @DarthVader It's not clear to me how you define $E: W \to Fix(\phi)$. In what sense are you taking an ultralimit? | |
| Nov 24, 2021 at 18:21 | comment | added | Darth Vader | You are very welcome :). I posted this as an answer. | |
| Nov 24, 2021 at 18:21 | answer | added | Darth Vader | timeline score: 1 | |
| Nov 24, 2021 at 17:58 | comment | added | Andromeda | @DarthVader Alright! Thanks. You may want to write an answer. | |
| Nov 24, 2021 at 17:56 | comment | added | Darth Vader | That's what I had in mind | |
| Nov 24, 2021 at 17:52 | comment | added | Andromeda | @DarthVader Ah okay, I think I see what you are referring to! The word non-principal ultrafilter threw me off (I'm not really familiar with filters). But basically you mean something like take a limit point (for the point $\sigma$-weak topology as Hamana refers to it) of the sequence of averages to construct the map $E$? Thanks again! | |
| Nov 24, 2021 at 17:48 | comment | added | Darth Vader | The conditional expectation defined here is same as Hamana's approach in "injective operator systems" paper. There he defines the "minimal projections" by looking at similar limits. Probably Paulsen's book also has similar approaches in various proofs- but I will have to check. | |
| Nov 24, 2021 at 17:46 | comment | added | Andromeda | @DarthVader Thanks! I'm definitely not familiar with such an approach. Do you have a reference for this? You may also want to write an answer. | |
| Nov 24, 2021 at 17:42 | comment | added | Darth Vader | If $W$ is an operator system, and $\phi$ a ucp map on $W$, then there exists a "conditional expectation" from W to Fix($\phi$). Define $E: W \rightarrow Fix(\phi)$ by $E(w)= \lim_{N \rightarrow \omega} \frac{1}{N} \sum_{n=1}^N \phi^n(w)$, where $\omega$ is a nonprincipal ultrafilter. As $W$ is injective, so is $Fix(\phi)$, by composition of "conditional expectations". | |
| Nov 24, 2021 at 17:32 | comment | added | Andromeda | @DarthVader Why is $\operatorname{Fix}(\varphi)$ injective? | |
| Nov 24, 2021 at 16:52 | comment | added | Darth Vader | Probably I'm missing something- but isn't Fix($\phi$), the set of fixed points of $\phi$, an injective operator system contained in W, that contains $\kappa(V)$? By minimality of $W$, we then have Fix($\phi$)$=W$, proving rigidity. | |
| Nov 23, 2021 at 14:25 | history | edited | Andromeda | CC BY-SA 4.0 |
added 7 characters in body
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| Nov 23, 2021 at 10:59 | history | asked | Andromeda | CC BY-SA 4.0 |