Yet Another Description of the Connes–Higson Functor

Suppose that $A$ and $B$ are $C^{*}$-algebras, $A$ is separable, and $B$ is stable. The elements of the group $E_{1}(A,B)$ in Connes–Higson $E$-theory are represented by $*$-homomorphisms from the suspension of $A$ to the asymptotic algebra $\mathfrak AB$. In the paper, an endofunctor $\mathfrak M$ in the category of $C^{*}$-algebras is constructed and a set of special homotopy classes of $*$-homomorphisms from $A$ to $\mathfrak{MA}B$ is defined so that this set endowed with the natural structure of an Abelian group coincides with $E_{1}(A,B)$.