A Composition Formula for Asymptotic Morphisms

For graded $C^{\ast }$-algebras $A$ and $B$, we construct a semigroup ${\cal AP}(A,B)$outofasymptoticpairs\mathrm{.}Thissemigroupissimilartothesemigroup$\Psi(A,B)$ofunboundedKK-modulesdefinedbyBaajandJulgandthereisamap$\Psi(A,B) \to {\cal AP}(A,B)$when$B$ is stable. Furthermore, there is a natural semigroup homomorphism $\mathcal{A}\mathcal{P}(A,B)\to E(A,B)$, where $E(A,B)$ is the E-theory group. We denote the image of this map $E^{\mathrm{\prime }}(A,B)$ and prove both that $E^{\mathrm{\prime }}(A,B)$ 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 $E^{\mathrm{\prime }}$ 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.