$E$-Theory is a Special Case of $KK$-Theory

Abstract

Let A and B be C*-algebras, with A separable and Bσ-unital and stable. It is shown that there are natural isomorphisms E ( A , B ) = K K ( S A , Q ( B ) ) = [ S A , Q ( B ) ⊗ K ] , where SA=C0(0, 1) ⊗ A, […, …] 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 of an infinite-dimensional separable Hilbert space. 2000 Mathematics Subject Classification 19K35, 19K33, 46M15.