\(K\)-homology, asymptotic representations, and unsuspended \(E\)-theory
Let \(A\), \(B\) be separable \(C^*\)-algebras, \(SA\) be the suspension of \(A\), \(E(A,B)\) be Connes/Higson bivariant homology theory (the elements of \(E(A,B)=[[SA\otimes\text{Comp};SB\otimes\text{Comp}]]\) are taken to be homotopy classes of so named asymptotic monomorphisms from \(SA\otimes K\) to \(SB\otimes K\)). The main purpose of this paper is to prove that \(E(A,B)=[[A\otimes\text{Comp};B\otimes\text{Comp}]]\) for some \(A\) and all \(B\). The class of those \(A\) include the dimension-drop intervals \[ A_n=\{f\in C([0,1];L(n;\mathbb{C}))\mid f(0), f(1)\text{ are scalar}\}. \] In this case, there is no distinctions between homomorphisms and asymptotic morphisms, and so these results are used in the problem of classification of certain \(C^*\)-algebras. The dimension-drop interval may also be used to describe \(K_*\)-groups and \(KK\)-groups: 1) for \(B\) separable \(K_0(B;\mathbb{Z}/n)\cong[[A_n;B\otimes\text{Comp}]]\), 2) for a locally compact space \(X\) with \(X^+\) having the homotopy type of a finite CW-complex \(K_1(X;\mathbb{Z}/n)=K^1(C_0(X);\mathbb{Z}/n)\cong[[C_0(X),A_n\otimes\text{Comp}]]\), 3) for \(A\) and \(B\) separable and \(A\) nuclear \(KK(A,B;\mathbb{Z}/n)\cong[[A_n\otimes A,B\otimes\text{Comp}]]\).
scientific article; zbMATH DE number 763974