K-Homology, Asymptotic Representations, and Unsuspended E-Theory

Connes and Higson defined a bivariant homology theory E(A, B) for separable C*-algebras. The elements of E(A, B) are taken to be homotopy classes of asymptotic momorphisms from SA ⊗ K to SB ⊗ K. In symbols E(A, B) ≅ [[SA, SB ⊗ K]]. If A is K-nuclear then E-theory agrees with Kasparov′s bivariant K-theory. We show that, in many cases, one need not take suspensions to calculate the E-theory group E(A, B). For many A, we show E(A, B) ≅ [[A, B ⊗ K]] for all B. Among the A for which this is true are C0(X\{pt}) for X with the homotopy type of a finite, connected CW complex. This gives a concrete realization of K-homology, related to the Brown-Douglas-Fillmore description. For example, K0(X) arises as asymptotic representations, K0(X) ≅ [[C0(X), K]]. Other A for which our isomorphic holds include the nonunital dimension-drop intervals. In this case, there is no distinction between ∗-homomorphisms and asymptotic morphisms so we have succeeded in classifying all ∗-homomorphisms from a dimension-drop interval to a stable C*-algebra. This was subsequently used by Elliott (J. Reine Angew. Math.443 (1993), 179-219) in the classification of certain C*-algebras. The dimension-drop interval may also be used to describe K*(X; Z/n) in terms of paths of asymptotic representations of C0(X).