Timeline for Is a NC sphere a (one point) compactification of a NC plane?
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| Jul 29 at 13:32 | comment | added | David Gao | I’ve added a proof that those relations generate $C_0(\mathbb{R}^2)$ as an answer. (Tangentially, the proof also shows that the universal unital algebra subject to those relations is indeed the algebra of the sphere.) As for your later follow-up comment, again, I’m not an expert in these quantum deformations. If you have something more specific in mind (namely, you can’t just say perturbing something, you have to define these deformations in a more rigorous way for me), maybe I can work with it. As of now, the most I can do are all contained in my answers here and to the linked question. | |
| Jul 29 at 13:20 | answer | added | David Gao | timeline score: 4 | |
| Jul 29 at 12:25 | comment | added | Ali Taghavi | @DavidGao I mean: Pick a classic non compact space $Y$ then one point compactify to $X$ then perturbe $X$ to $X_q$ in the virtual word which commutativity is forbidden then try to realize $X_q$ as a minimal unitization. This is my concern. | |
| Jul 29 at 12:13 | comment | added | Ali Taghavi | @DavidGao Thank you for your attention to my question. Ok we can talk later if you have no time just now. Since last night I realy wonder if there would be a unital perturbation whose realization as minimal unitization of a non unital perturbation would produce some interesting stories. We possibly talk later | |
| Jul 29 at 12:05 | comment | added | David Gao | I’m not sure what you’re talking about. $a, b$ are functions on $\mathbb{R}^2$, not coordinate transform maps, and they both send the point at infinity to $0$, as is required for them to be even in $C_0(\mathbb{R}^2)$, not to the equator. I’ll write a rigorous proof later. I don’t have time at the moment, but it’s easier to work with the one-point compactification and first show its one-point compactification is the sphere. | |
| Jul 29 at 10:43 | comment | added | Ali Taghavi | Note that $C_0(R^2)$ correspond to 1 point compactification but the fraction you used compactify the infinity to ${(x,y,o)\in S^2$$ | |
| Jul 29 at 10:35 | comment | added | Ali Taghavi | @DavidGao So Please write a rigorous proof that they generate $C_0(R^2)$ | |
| Jul 29 at 10:32 | comment | added | Ali Taghavi | @DavidGao Yes you did. But my question is that why these relations generate $C_0(R^2)$? the term $\frac{1}{x^2+y^2+1}$ you idicated to remind of "Poincare compactification" which is NOT a 1 point compactification. In this case the equator of sphere play the role of infinity(Not a unique point $\infty$) | |
| Jul 29 at 3:53 | comment | added | David Gao | What did you mean by “do relations you mentioned work and generate the algebra”? I already said the universal algebra subject to those relations is $C_0(\mathbb{R}^2)$, didn’t I? | |
| Jul 29 at 3:52 | comment | added | David Gao | I wrote $a(1-a)$ for the sake of making the analogies to the relations defining $A_q$ clear. It’s not really an issue as you can always pass to the unitization and work there. As long as it ends up landing in the original algebra, it doesn’t matter. | |
| Jul 29 at 3:49 | comment | added | David Gao | I’m not sure about any of these. I’d suggest you read the literature on various quantum algebras to find out. Again, I’m not an expert in these topics. I’m simply dealing with the definition of quantum spheres as laid before me. I’m not even sure how would you rigorously define “non-commutative deformation” or “perturbation” of classical functional algebras in full generality. | |
| Jul 28 at 20:30 | comment | added | Ali Taghavi | @DavidGao Any way it would be interesting to have a perturbation of a compact space which is not minimal compactification of any non compact perturbation. can one imagin a possible example? | |
| Jul 28 at 19:58 | comment | added | Ali Taghavi | I remember that descriptiin of classical and NC torus is inspir3d by Fourier series periodic functions in two variabkes. So how do you describe NC non compact plane? | |
| Jul 28 at 19:48 | comment | added | Ali Taghavi | @DavidGao I guess you mean$a-a^2$ not $a(1-a)$? BTW do relations you mentioned work and generate the algebra? | |
| Jul 28 at 19:10 | comment | added | Ali Taghavi | @DavidGao thank you. what can be said about the generalization? The last part of the question? Is there a perturbation $A_q$ of a classical space X which is a compactification of a classical Y such that no perturbation of $C_0(Y)$ can be found whose unitization is $A_q$? | |
| Jul 28 at 18:55 | comment | added | David Gao | … universal (non-unital) $C^\ast$-algebra generated by $a,b$ subject to relations listed in the linked post, it seems we can say $B_q$ is a deformation of $C_0(\mathbb{R}^2)$ in some suitable sense, and indeed $A_q$ is the unitization of $B_q$. | |
| Jul 28 at 18:52 | comment | added | David Gao | $C_0(\mathbb{R}^2)$ is generated by two functions $a = \frac{1}{1+x^2+y^2}$ and $b = \frac{x+iy}{1+x^2+y^2}$. They are subject to the relations $a=a^\ast$, $ab=ba$, $ab^\ast=b^\ast a$, $bb^\ast=b^\ast b=a(1-a)$, and it is not hard to verify that $C_0(\mathbb{R}^2)$ is exactly the universal (non-unital) $C^\ast$-algebra generated by $a,b$ subject to those relations. If you introduce a parameter $q$ and deform these relations by $q$, you can get exactly the defining relations of $A_q$ in the linked post, with the only difference being we’re no longer requiring a unit. If we let $B_q$ be the… | |
| Jul 28 at 18:36 | history | edited | Ali Taghavi |
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| Jul 28 at 18:29 | history | asked | Ali Taghavi | CC BY-SA 4.0 |