4
$\begingroup$

The Gelfand--Naimark theorem tells us to regard noncommutative $C^*$-algebras as "noncommutative function spaces". In that spirit $K$-theory the Grothendieck group of "noncommutative vector bundles" and $K$-homology the "homotopy classes of differential operators".

I want to see what $KK(A,B)$ is from this point of view, for two $C^*$-algebras $A,B$. To make my question more concrete: Is there a geometric/topological construction of $KK(C(X),C(Y))$ for $X,Y$ two (compact) topological spaces/ differential manifolds? If such a thing exists - then what is the Kasparov pairing in this setting?

Improve this question
$\endgroup$
1
  • 2
    $\begingroup$ Exercise 19.9.6 of Blackadar's text K-theory for Operator algebras seems to have what you are looking for when Y is a smooth manifold. Bruce cites the 1984 paper of Connes and Skandalis for that exercise. $\endgroup$
    – Caleb Eckhardt
    Commented Jun 6, 2022 at 16:08

0

Reset to default

You must log in to answer this question.