On the asymptotic homotopy type of inductive limitC *-algebras

Abstract

Let X, Y be compact, connected, metrisable spaces with base points Xo, Yo and let denote the compact operators. It is shown that Co(X\xo)| is asymptotically homotopic (or shape equivalent) to Co(Y~yo) @ Yi r if and only if X and Y have isomorphic K-groups. Similar results are obtained for certain inductive limits of nuclear C*-algebras. Let ~r denote the category whose objects are all the separable C*-algebras and whose set of morphisms from A to B, denoted [[A, B]], consists of homotopy classes of asymptotic morphisms. The construction of this category is due to Connes and Higson [CH], who defined a bivariant homology theory E(A, B) = lISA, SB | ~ff]] and have shown how to define the intersection product for arbitrary extensions of separable C*-algebras. If A is K-nuclear then E-theory agrees with Kasparov's bivariant K-theory [K]. On the other hand the asymptotic homotopy category d appears to be the "right" framework for the homotopy theory of separable C*-algebras. This point of view is supported by results in [H; CH; D; CHI; CK; D1]. For instance we have shown in [D1] that asymptotic homotopy is equivalent to a strong shape theory and hence is intimately related to the shape theories of [EK1] and [B] which were intended as homotopy theories for noncommutative singular spaces. In particular it turned out that two separable C*-algebras are shape equivalent if and only if they are asymptotically homotopic i.e. isomorphic in ~r The isomorphism class in sr of a separable C*-algebra A is called the asymptotic homotopy type of A. In this note we exhibit large classes of (projeetionless) stable, nuclear C*algebras whose asymptotic homotopy type is determined by K-theoretical data (Theorem 6). This is done via a suspension isomorphism