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
18 events
| when toggle format | what | by | license | comment | |
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| Oct 26, 2018 at 13:23 | comment | added | Nik Weaver | @YCor: yes, in my example after you factor out $U(1)$ the classes of the $U_{f_n}$ still fail to converge in the quotient of the norm topology. | |
| Oct 26, 2018 at 6:21 | vote | accept | Transcendental | ||
| Oct 26, 2018 at 4:30 | comment | added | YCor | @Nik Oh, I forgot (and you also) to mod out by the circle group, but it's essential since it indeed yields an obstruction to lifting [this is the OP's comment to your answer]. This kind of things is why I so much insist on giving basic context! Hence, every action as given yields a map $G\to U(H)/U(1)$ and certainly the question is more interesting if one asks about continuity of this map with $U(H)/U(1)$ endowed with quotient of the norm topology. Does your proof adapt to showing this? | |
| Oct 26, 2018 at 1:05 | comment | added | Nik Weaver | @Transcendental: yes, you're right. (My parenthetical statement is correct as it stands.) | |
| Oct 26, 2018 at 0:51 | comment | added | Transcendental | @NikWeaver: Shouldn’t it be $ \mathbb{U}(\mathcal{H}) $ modulo the circle group instead? | |
| Oct 26, 2018 at 0:45 | comment | added | Nik Weaver | @YCor: that is correct, $U(H)$ is the automorphisms group of $K(H)$. ($K(H)$ has only one irrep up to unitary equivalence.) | |
| Oct 25, 2018 at 23:50 | answer | added | Nik Weaver | timeline score: 3 | |
| Oct 25, 2018 at 22:02 | comment | added | YCor | Thanks! My other question was, regardless of the topology: do you already know what is the automorphism group of $K(H)$ (as a $C^\ast$-algebra), namely, is it reduced to $U(H)$? That is, do you know the answer of your question to be positive when $G$ is discrete? | |
| Oct 25, 2018 at 21:58 | comment | added | Transcendental | @YCor: I apologize for any confusion that I may have caused you. I’ve clarified in my post what I mean by a group action on a $ C^{\ast} $-algebra. | |
| Oct 25, 2018 at 21:58 | history | edited | YCor |
edited tags
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| Oct 25, 2018 at 21:56 | history | edited | Transcendental | CC BY-SA 4.0 |
I gave clarification for what I meant by a group action on a $ C^{\ast} $-algebra.
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| Oct 25, 2018 at 20:08 | comment | added | YCor | @YemonChoi please don't impugn motives. It was pretty clear from my 1st comment that I expected the OP to have implicit hypotheses, but unlike you, I wasn't sure which ones. | |
| Oct 25, 2018 at 20:06 | comment | added | Yemon Choi | @YCor The fact that the answer is "obviously not" if we work in the generality mentioned in your first comment seems a pretty strong indication to me that the OP did not intend to work in this generality, unless you have a very low opinion of the OP's ability to check examples. | |
| Oct 25, 2018 at 19:59 | comment | added | Yemon Choi | Analogy: I am quite happy talking about locally compact groups and to read other people talking about locally compact groups and have come to accept the language of "quasi-compact". | |
| Oct 25, 2018 at 19:58 | comment | added | Yemon Choi | I am coming late to this, but as someone who is merely a practising functional analyst, it seems clear to me from the context of the question that the OP is asking about homomorphisms $\alpha: G \to {\rm Aut}{\mathbb K}({\mathcal H})$ where the automorphisms are in the category of ${\rm C}^*$-algebras -- which makes them isometric, $*$-preserving, non-degenerate, etc etc. | |
| Oct 25, 2018 at 19:51 | comment | added | Matthew Daws | I have deleted my comments as they are seemingly not helpful | |
| Oct 25, 2018 at 18:50 | comment | added | YCor | Obviously not, if the action of $G$ on $K(H)$ does not preserve the multiplication (i.e., is not by non-unital algebra automorphisms). Also it should commute with the $\ast$-involution. Are these missing hypotheses? The answer should be clear for $G=\mathbf{Z}$, to start with. | |
| Oct 25, 2018 at 18:23 | history | asked | Transcendental | CC BY-SA 4.0 |