A Composition Formula for Asymptotic Morphisms

TL;DR

This paper introduces a new composition formula for asymptotic morphisms in the setting of graded C*-algebras, connecting asymptotic pairs, KK-theory, and E-theory with a focus on composition products.

Contribution

It constructs a semigroup of asymptotic pairs, relates it to KK-modules and E-theory, and provides a new formula for composition products under specific hypotheses.

Findings

Defined a semigroup ${ m AP}(A,B)$ of asymptotic pairs.

Established a homomorphism from ${ m AP}(A,B)$ to E-theory groups.

Derived a composition formula for the product on $E'$ under certain conditions.

Abstract

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.