KK-theory

Definition

Let $A$ be a separable C*-algebra and $B$ a $\sigma$-unital C*-algebra, both with the trivial grading. For a Hilbert C*-module $E$ over $B$, let $\mathcal{L}(E)$ be the C*-algebra of adjointable operators on $E$. Let $\mathcal{K}(E)$ be its ideal of compact operators, the closed span of the operators $\theta_{x, y}(z) = x \langle y, z \rangle$. A Kasparov $(A, B)$-module is a triple $(E, \varphi, F)$ where:

1. $E = E^{(0)} \oplus E^{(1)}$ is a countably generated $\Z/2$-graded Hilbert $B$-module;
2. $\varphi \colon A \to \mathcal{L}(E)$ is a $*$-homomorphism into the even operators, those that preserve $E^{(0)}$ and $E^{(1)}$;
3. $F \in \mathcal{L}(E)$ is odd, that is, it maps $E^{(0)}$ to $E^{(1)}$ and $E^{(1)}$ to $E^{(0)}$;
4. for all $a \in A$, the operators $(F^2 - 1)\varphi(a)$, $(F - F^*)\varphi(a)$ and $F \varphi(a) - \varphi(a) F$ lie in $\mathcal{K}(E)$.

Two Kasparov $(A, B)$-modules are homotopic if they are the images, under evaluation at $0$ and at $1$, of a Kasparov $(A, C([0, 1], B))$-module. The group $KK(A, B)$ is the set of homotopy classes of Kasparov $(A, B)$-modules, with addition given by the direct sum of modules.