A note on asymptotic homomorphisms
The aim of this paper is to introduce homology and cohomology theories for separable \(C^*\)-algebras which satisfy excision but are not periodic. The construction of these theories is based on the notion of asymptotic homomorphisms of \(C^*\)-algebras due to Connes and Higson. The author first reviews the basic properties of asymptotic homomorphisms and homotopies of asymptotic homomorphisms. Next, for separable \(C^*\)- algebras \(A\) and \(B\) he considers the set \(A,B\) of homotopy classes of asymptotic homomorphisms from \(A\) to \(B\) and defines \(\{\{A,B\}\}_ q\) as \(\lim_ k [[S^{k + q} A, S^ kB]]\), where \(S^ \ell\) denotes the \(\ell\)-fold suspension. Next, he constructs a Puppe sequence for the functor \(C, _ -\) and proves that for a short exact sequence \[ 0 \to J \to A@>p>> B \to 0 \] the suspension \(SJ\) is isomorphic to the mapping cone \(C_ p\) in the homotopy category of asymptotic homomorphisms. These results imply that if \(A\) is a separable \(C^*\)-algebra, then \(\{\{A, _ -\}\}_ *\) is a homology theory and \(\{\{_ -,A\}\}_ *\) is a cohomology theory. For a stable separable \(C^*\)-algebra \(B\) the group \(\{\{A,B\}\}\) is shown to coincide with the \(E\)-group \(E(A,B)\) of Connes and Higson. Finally, it is shown that if \(X\) is a compact ANR, then \(\{\{\mathbb{C}, C(X)\}\}_ *\) is the stable homotopy of \(X\) and \(\{\{C(X), \mathbb{C}\}\}_ *\) is stable cohomotopy, so the theories are non- periodic in general.
scientific article; zbMATH DE number 720848