Timeline for Strongly Continuous Group Actions on the $ C^{\ast} $-Algebra of Compact Operators on a Hilbert Space
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 1, 2018 at 0:16 | comment | added | Transcendental | @NikWeaver: This may sound like a silly question, Nik. Do you happen to know if every strongly continuous action of $ G $ on $ \mathbb{K}(\mathcal{H}) $ extends to a strongly continuous action of $ G $ on $ \mathbb{B}(\mathcal{H}) $? I’m assuming $ G $ to be an arbitrary locally compact Hausdorff group. | |
| Oct 26, 2018 at 7:59 | comment | added | Matthew Daws | I wonder, if I may, I could make a comment that as I see it, there are two separate things going on here. One is the issue of projective representations. The other is an issue of topology. If we give $U(H)$ the SOT topology, then in Nik's example, we do have that $U_{f_n}\rightarrow 1$. I believe that if you equip $U(H)$ with the SOT then only the cohomological obstruction is left. I don't know how to characterise the norm-continuous case (perhaps look at actions on all of $B(H)$?) | |
| Oct 26, 2018 at 6:22 | comment | added | Transcendental | Yes, we always get a projective representation. | |
| Oct 26, 2018 at 6:21 | vote | accept | Transcendental | ||
| Oct 26, 2018 at 3:03 | comment | added | Nik Weaver | Right, I remember that now. You always get a projective unitary representation though, don't you? | |
| Oct 26, 2018 at 0:44 | comment | added | Transcendental | Thanks for your counterexample, Nik! Actually, I just found out that my question has a negative answer by way of what’s called the “Mackey obstruction”. It’s an element of $ {H^{2}}(G,\mathbf{T}) $ associated to every strongly continuous action $ \alpha $ of $ G $ on $ \mathbb{K}(\mathcal{H}) $ with the property that if it isn’t trivial, then $ \alpha $ can’t be implemented by even an algebraic homomorphism from $ G $ to $ \mathbb{U}(\mathcal{H}) $, much less a norm-continuous one. | |
| Oct 25, 2018 at 23:50 | history | answered | Nik Weaver | CC BY-SA 4.0 |