E-theory

Summary

E-theory, introduced by Connes and Higson, assigns an abelian group $E(A, B)$ to each pair of separable C*-algebras, built from asymptotic morphisms. Like KK-theory, it has a composition product and contains K-theory and K-homology. Unlike KK-theory, it is half-exact in both variables for every extension of C*-algebras, not only for semisplit ones. There is a natural map $KK(A, B) \to E(A, B)$, and it is an isomorphism when $A$ is nuclear.