Timeline for Direct sums of operator spaces
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 14, 2023 at 10:50 | answer | added | Matthew Daws | timeline score: 4 | |
| Sep 12, 2023 at 19:08 | comment | added | J. De Ro | @Matthew Daws Is there any follow-up on this? I'm interested in the same thing, so maybe you discovered some useful references? | |
| May 7, 2019 at 3:42 | comment | added | Yemon Choi | All this is related to the POV where the dual space functor (let's say for $\newcommand{\BSp}{\sf BSp}\BSp$ but probably this works for OpSp as well) is viewed as $D: \BSp \to \BSp^{op}$ with corresponding $D^{op}:\BSp^{op}\to\BSp$, and then $D$ is left adjoint to $D^{op}$ but $D^{op}$ is not left adjoint to $D$. Now left adjoints preserve colimits, and since binary coproduct in $\BSp$ is $\oplus_1$ while binary product is $\oplus$, you get $D(E\oplus_1 F) = (DE)\oplus (DF)$. But $D^{op}$ need not send colimits in $\BSp^{op}$ (limits in $\BSp$) to colimits in $\BSp$. | |
| May 7, 2019 at 3:27 | comment | added | Yemon Choi | This is illustrated by considering the category of Banach spaces and linear contractions, when the coproduct of countably many copies of the ground field is $\ell_1$ while the product of countably many copies of the ground field is $\ell_\infty$, and we have $(\ell_1)^*\cong\ell_\infty$ but $(\ell_\infty)^* \not\cong \ell_1$ | |
| May 7, 2019 at 3:25 | comment | added | Yemon Choi | belated comment about the last part: there is a sort of meta-argument why one can expect a universal-property proof of the first isomorphism but might have to work harder for the second one. Namely, duals of coproducts will be products of the duals, but duals of products need not be coproducts of the duals | |
| May 2, 2019 at 17:26 | history | asked | Matthew Daws | CC BY-SA 4.0 |