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I am looking for a detailed proof of the analytic index for families (of geometrically defined Dirac operators is good enough and assuming the existence of the kernel bundle) is multiplicative. Any reference, including papers, lecture notes, is very welcome.

If I am not mistaken, the proof of the multiplicativity of the analytic index for single elliptic operator can be found, for example, in page 249 of Lawson-Michelsohn's Spin geometry. Or is the proof of that for the analytic index for families is just a generalization of the single operator case?

I just realized I asked a similar question about 6 years ago

Multiplicativity of the analytic index (or of kernel bundle)

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    $\begingroup$ 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? $\endgroup$
    – Paul Siegel
    Commented Jul 12, 2022 at 18:15
  • $\begingroup$ @PaulSiegel Thanks. I will take a look at Blackadar's book. I was expecting a proof entirely within K-theory for some reason. $\endgroup$
    – Ho Man-Ho
    Commented Jul 13, 2022 at 5:25
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    $\begingroup$ 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. $\endgroup$
    – Paul Siegel
    Commented Jul 13, 2022 at 12:47
  • $\begingroup$ @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. $\endgroup$
    – Ho Man-Ho
    Commented Jul 14, 2022 at 8:19

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