Timeline for When forgetting structure doesn't matter
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Apr 30, 2022 at 18:41 | history | made wiki | Post Made Community Wiki by Stefan Kohl♦ | ||
| Apr 30, 2022 at 18:21 | comment | added | Yemon Choi | If your work in the respective categories where morphisms are required to have norm at most 1 then we might get something like your original claim, via the Vidav-Palmer theorem; but I am currently occupied with other tasks and haven't sat down to work out the details. | |
| Apr 30, 2022 at 18:16 | comment | added | Yemon Choi | One of my points is that you haven't specified if your homomorphisms between the Banach algebras that happen to have suitable involutions are supposed to be star homomorphisms. In the category of Cstar algebras you want the morphisms to be star homomorphisms. So I think this example might need to be taken back to the drawing board. | |
| Apr 30, 2022 at 16:27 | comment | added | Qfwfq | @YemonChoi: you're right, I've been a bit sloppy. I just remembered that if a Banach algebra has an involution that makes it into a C*-algebra, then such involution is unique. I was assuming the morphisms on both sides are just continuous homomorphisms. Perhaps it's a bit tautologous (many things in category theory are tautologous in sense though..). If you know some relevant functional analysis to make my answer more interesting you're welcome to do so. :) | |
| Apr 30, 2022 at 14:06 | comment | added | Yemon Choi | For that matter: is the category on the left what used to be called (the category of) $B^*$-algebras, i.e. Banach star-algebras with an involution satisfying the Cstar condition? If Cstar algebras here mean closed star subalgebras of some B(H), then what are the morphism? | |
| Apr 30, 2022 at 14:05 | comment | added | Yemon Choi | Can you describe the category on the right in more detail, in particular what the morphisms are? I am having trouble fleshing out this example without making it tautologous. | |
| Apr 30, 2022 at 13:13 | history | answered | Qfwfq | CC BY-SA 4.0 |