Timeline for Necessary conditions for $K_0(I_x\bigotimes A)$ to be the trivial group
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| Feb 20, 2023 at 20:23 | comment | added | Jamie Gabe | The essence of what I meant is that if you want to check that $K_0(I_x \otimes A)$ vanishes, it is necessary and sufficient to see how $K_i(I_x)$ and $K_j(A)$ relate to each other wrt $\otimes$ and $Tor$ (as described above). You can find the Künneth theorem in [Schochet, Claude Topological methods for C∗-algebras. II. Geometric resolutions and the Künneth formula. Pacific J. Math. 98 (1982), no. 2, 443–458] | |
| Feb 20, 2023 at 19:35 | comment | added | Sanae Kochiya | Thank for your input. I am sorry to tell you I have no backgrounds related to homology and cannot understand how theorem could help. May I ask for a link to references relevant to the Künneth theorem, or its applications? | |
| Feb 18, 2023 at 13:37 | comment | added | Jamie Gabe | I'm not sure if this is along the lines of what you're looking for in question 1: If $A$ is separable and $X$ is second countable, then the Künneth theorem (which holds when one $C^*$-algebra is abelian (or more generally, is nuclear and satisfies the UCT)) implies that $K_0(I_x \otimes A)=0$ if and only if the following four groups vanish: $K_0(I_x) \otimes K_0(A)$, $K_1(I_x)\otimes K_1(A)$, $Tor(K_0(I_x), K_1(A))$, and $Tor(K_1(I_x), K_0(A))$. I expect you can get this for arbitrary $A$ and $X$ as well by limiting over separable $C^*$-subalgebras. | |
| Feb 13, 2023 at 2:40 | history | asked | Sanae Kochiya | CC BY-SA 4.0 |