A $KK$-like picture for $E$-theory of $C^*$-algebras
Let $A$, $B$ be separable $C^*$-algebras, $B$ stable and $\sigma $-unital. Elements of the $E$-theory group $E(A,B)$ are represented by asymptotic homomorphisms from the second suspension of $A$ to $B$. Our aim is to represent these elements by (families of) maps from $A$ itself to $B$. We have to pay for that by allowing these maps to be even further from $*$-homomorphisms. We prove that $E(A,B)$ can be represented by pairs $(\varphi ^+,\varphi ^-)$ of maps from $A$ to $B$ which are not necessarily asymptotic homomorphisms, but have the same deficiency from being ones. Not surprisingly, such pairs of maps can be viewed as pairs of asymptotic homomorphisms from some $C^*$-algebra $C$ that surjects onto $A$, and the two maps in a pair should agree on the kernel of this surjection. We give examples of full surjections $C\to A$, i.e. those for which all classes in $E(A,B)$ can be obtained from pairs of asymptotic homomorphisms from $C$.