E-theory is a special case of KK-theory

Abstract

Let A and B be $C^*$-algebras, A separable, and B $\sigma$-unital and stable. It is shown that there are natural isomorphisms $E(A,B)=KK(SA,Q(B))=[SA,Q(B)\otimes K]$, where $SA=C_0(0,1)\otimes A$, $[\cdot,\cdot]$ denotes the set of homotopy classes of *-homomorphisms, $Q(B) = M(B)/B$ is the generalized Calkin algebra and K denotes the $C^*$-algebra of compact operators.