Timeline for When do completely positive maps have a closed image?
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 24, 2020 at 8:56 | vote | accept | Diego Martinez | ||
| Jan 23, 2020 at 17:50 | answer | added | Yemon Choi | timeline score: 5 | |
| Jan 23, 2020 at 17:44 | comment | added | Robert Israel | Hmm, I take that back. I think it is not enough for $V$ to have closed range. | |
| Jan 23, 2020 at 17:27 | comment | added | Yemon Choi | @RobertIsrael If the cp map sends 1 to 1 then as Diego points out, the V constructed in Stinespring's representation is an isometry. But I think that by choosing suitable integral operators on C[0,1] one can get positive operators that send 1 to 1 yet do not have closed range. | |
| Jan 23, 2020 at 16:59 | comment | added | Diego Martinez | @RobertIsrael if you ever find the time and energy to provide a sketch of the proof that'd be great, because that'd mean that unital cp maps, for instance, do have closed range. Thank you as well | |
| Jan 23, 2020 at 16:58 | comment | added | Diego Martinez | @YemonChoi that, erm, makes a lot of sense. What I meant was whether injective cp maps of norm 1 are isometric, but that is also false. Thank you for pointing it out. | |
| Jan 23, 2020 at 16:43 | comment | added | Robert Israel | In the Stinespring case, you want $\mathcal B = B(H)$ where $H$ is a Hilbert space, $\pi$ is a *-homomorphism into $B(K)$ for some Hilbert space $K$, and $V: K \to H$. I think in this case you should have closed image if $V$ has closed range. | |
| Jan 23, 2020 at 16:15 | comment | added | Yemon Choi | Based on the uiquity of counterexamples (i.e. cp maps with non-closed range) I don't think "under which conditions to cp maps have closed range" is sufficiently focused to make a good MO question: they have closed range when, erm, they have closed range. | |
| Jan 23, 2020 at 16:14 | comment | added | Yemon Choi | You should be able to easily find counterexamples for commutative A and B, when cp is the same as usual positivity by Naimark's theorem (see Effros-Ruan or Paulsen's book). I think this is a useful exercise for you to figure out. | |
| Jan 23, 2020 at 16:00 | review | First posts | |||
| Jan 23, 2020 at 18:20 | |||||
| Jan 23, 2020 at 15:58 | history | asked | Diego Martinez | CC BY-SA 4.0 |