A Composition Formula for Asymptotic Morphisms

For graded $C^*$-algebras $A$ and $B$, we construct a semigroup ${\cal AP}(A,B)$ out of asymptotic pairs. This semigroup is similar to the semigroup $\Psi(A,B)$ of unbounded KK-modules defined by Baaj and Julg and there is a map $\Psi(A,B) \to {\cal AP}(A,B)$ when $B$ is stable. Furthermore, there is a natural semigroup homomorphism ${\cal AP}(A,B) \to E(A,B)$, where $E(A,B)$ is the E-theory group. We denote the image of this map $E’(A,B)$ and prove both that $E’(A,B)$ is a group and that the composition product of E-theory specializes to a composition product on these subgroups. Our main result is a formula for the composition product on $E’$ under certain operator-theoretic hypotheses about the asymptotic pairs being composed. This result is complementary to known results about the Kasparov product of unbounded KK-modules.