Timeline for A proof that the analytic index for families is multiplicative
Current License: CC BY-SA 4.0
5 events
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| Jul 14, 2022 at 8:19 | comment | added | Ho Man-Ho | @PaulSiegel Yes, you are right that KK-theory is a very convenient tool. Unfortunately, my training never involved KK-theory and operator algebra. I now think even if I want to work out a proof for the family case for Dirac operators entirely within K-theory, probably I still need to learn KK-theory to get the idea. | |
| Jul 13, 2022 at 12:47 | comment | added | Paul Siegel | Well, the functional analysis is bound to be quite involved no matter how you proceed. KK-theory is convenient because it wraps up all of the hard analysis into a few nontrivial theorems, like the fact that the Kasparov product exists and is associative, and that the KK-class of a product is the product of KK-classes. But it should still be possible to work out the analysis by hand if desired. | |
| Jul 13, 2022 at 5:25 | comment | added | Ho Man-Ho | @PaulSiegel Thanks. I will take a look at Blackadar's book. I was expecting a proof entirely within K-theory for some reason. | |
| Jul 12, 2022 at 18:15 | comment | added | Paul Siegel | Well, I don't have a reference handy, but this should be pretty straightforward using Kasparov products. A family of fiberwise elliptic operators determines a class in KK-theory, and the families index map is the Kasparov product of this KK-theory class with the K-theory class of a vector bundle over the base. But the KK-theory class of the product of two families of operators is the Kasparov product of the KK-theory classes of the families, so the result follows from associativity of Kasparov products. Maybe this is worked out in Blackadar's book? | |
| Jul 12, 2022 at 17:16 | history | asked | Ho Man-Ho | CC BY-SA 4.0 |