Timeline for Kasparov's Dirac element and the index map
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| S Mar 24, 2022 at 12:34 | history | suggested | The Amplitwist | CC BY-SA 4.0 |
fixed broken link to springerlink.com
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| Mar 24, 2022 at 8:36 | review | Suggested edits | |||
| S Mar 24, 2022 at 12:34 | |||||
| Oct 25, 2012 at 13:59 | answer | added | Alain Valette | timeline score: 1 | |
| Oct 25, 2012 at 7:35 | comment | added | Alain Valette | @Zhaoting: Thanks a lot for the clarification. I think however that you should write $K^n_G(X)$ rather than $K^G_n(X)$, because it is K-theory, i.e. a theory contravariant in $X$. | |
| Oct 25, 2012 at 4:41 | history | edited | Zhaoting Wei | CC BY-SA 3.0 |
added 43 characters in body
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| Oct 25, 2012 at 4:34 | history | edited | Zhaoting Wei | CC BY-SA 3.0 |
I defined KG(X) to be the K-theory of the cross product algebra.
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| Oct 25, 2012 at 3:34 | comment | added | Zhaoting Wei | @Alain: (Continued) The index map I am looking for is the same as that in Penington-Plymen paper. I apologize for the confusing I've made and I have made some modifications in the question body to define equivariant K-theory more clearly. Nevertheless it is really appreciated that you pointed out the problem. | |
| Oct 25, 2012 at 3:29 | comment | added | Zhaoting Wei | @Alain: Thank you for your comments! Sure I should be more clear about what I want. The second point you made may be because of the non-standard terminology I used. In fact for any Lie group $G$ I define $K^G_0(X)=K(C_r^*(G; C_0(X)))$ where $C_r^*(G; C_0(X))$ is the reduced cross product $C^*$-algebra. Hence the $K_*(C^*_r(G))$ is exactly $K^G_*(pt)$ in my question. As for $R(U)$, since $U$ is compact, $R(U)=K_0(C^*_r(U))$. We also notice that $C^*_r(U)$ is strong Morita equivalent to $C^*_r(G; C_0(G/U))$, hence $R(U)=K^G_0(G/U)$ in this context. The index map I am looking for is the same as | |
| Oct 24, 2012 at 23:05 | comment | added | Alain Valette | First, I think that you should be more specific about the properties you expect of an index map. Second, I checked the Penington-Plymen paper and there is no mention of the index map you indicate; their index map goes from the representation ring $R(U)$ to the K-theory $K_*(C^*_r(G))$ of the reduced $C^*$-algebra of $G$. | |
| Oct 24, 2012 at 20:24 | history | edited | Zhaoting Wei | CC BY-SA 3.0 |
In paragraph 3 I drop the assumtion that dim X is even.
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| Oct 24, 2012 at 17:32 | history | asked | Zhaoting Wei | CC BY-SA 3.0 |