Timeline for A coalgebra structure on compact operators
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Feb 9, 2015 at 18:33 | vote | accept | Ali Taghavi | ||
| Nov 9, 2014 at 20:34 | comment | added | Todd Trimble | @YemonChoi Yes, I think you are probably right. I composed my answer ignoring the topological aspects, which were brought more forcefully to attention by Ali's edit. So what he is really after no doubt deserves more attention. | |
| Nov 9, 2014 at 20:28 | comment | added | Yemon Choi | Todd, if we work in the symmetric monoidal category Bang of Banach spaces and maps of norm at most 1, then doesn't one have to tweak your initial remarks about hom(V,V) being self-dual? It seems one would need to do a bit more work to create a coalgebra structure on ${\rm End}_{\rm Bang}(\ell_2^n)$ a.k.a. $M_n({\bf C})$ with its usual norm. But perhaps I am missing something. | |
| Nov 9, 2014 at 20:09 | comment | added | Todd Trimble | @AliTaghavi Keeping in mind that I am not a functional analyst, the coalgebra comultiplication $\text{colim}\; M_n \to (\text{colim}\; M_n) \otimes (\text{colim}\; M_n)$ would be induced (via the universal property of colimits) from the cocone $M_n \to M_n \otimes M_n \to (\text{colim}\; M_n) \otimes (\text{colim}\; M_n)$ where $M_n \to \text{colim}\; M_n$ is the canonical inclusion. Here colimits = direct limits would be computed in a suitable TVS sense. Does that help? | |
| Nov 9, 2014 at 20:06 | comment | added | Ali Taghavi | could you please more explain on your last comment? and also your extension of comultiplication to the space of compact operators? | |
| Nov 9, 2014 at 20:00 | comment | added | Todd Trimble | Ali's recent edit inserted the word "topological", which might introduce some delicacy since there are various tensor products one might consider on TVS. But for any symmetric monoidal category $M$, the forgetful functor $\text{Coalg}(M) \to M$ still creates colimits. | |
| Nov 9, 2014 at 19:59 | comment | added | Ali Taghavi | thank you for your interesting answer. Could you please explain how we extend the comultiplication to the space of compact operators(The closure of the algebraic direct limit)? | |
| Nov 9, 2014 at 19:17 | history | undeleted | Todd Trimble | ||
| Nov 9, 2014 at 19:17 | history | edited | Todd Trimble | CC BY-SA 3.0 |
added 640 characters in body
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| Nov 9, 2014 at 18:50 | history | deleted | Todd Trimble | via Vote | |
| Nov 9, 2014 at 18:49 | history | answered | Todd Trimble | CC BY-SA 3.0 |