A note on asymptotic homomorphisms
type: Item
instance of: scholarly article
title: A note on asymptotic homomorphisms
zbMATH Open document ID: 0821.19003
DOI: 10.1007/BF00961401
published in: \(K\)-Theory
publication date: 1995-09-25T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 19K35
Mathematics Subject Classification ID: 46L80
Mathematics Subject Classification ID: 46M20
Mathematics Subject Classification ID: 46L85
zbMATH DE Number: 720848
zbMATH Keywords: homology for \(C^*\)-algebras
zbMATH Keywords: asymptotic homomorphisms
zbMATH Keywords: homotopy category
zbMATH Keywords: stable homotopy
zbMATH Keywords: stable cohomotopy
zbMATH Keywords: non-periodic
author: Marius Dadarlat
reviewed by: Andreas Cap
MaRDI profile type: Publication
cites work: The completely positive lifting problem for \(C^*\)-algebras
cites work: Categories of fractions and excision in KK-theory
cites work: THE OPERATORK-FUNCTOR AND EXTENSIONS OFC*-ALGEBRAS
cites work: Q3205815
cites work: The role of 𝐾-theory in noncommutative algebraic topology
cites work: Topological methods for \(C^*\)-algebras. III: Axiomatic homology
cites work: Q3983714
cites work: Shape theory and asymptotic morphisms for \(C^*\)-algebras
cites work: Generalized homology theories on compact metric spaces
cites work: An exact sequence from the nth to the (n-1)-st fundamental group
cites work: Shape theory for $C^*$-algebras.
full work available at URL: https://doi.org/10.1007/bf00961401
OpenAlex ID: W2084206071
description: scientific article; zbMATH DE number 720848