\(C^*\)-algebra homomorphisms and \(KK\)-theory
type: Item
instance of: scholarly article
title: \(C^*\)-algebra homomorphisms and \(KK\)-theory
zbMATH Open document ID: 0945.46049
DOI: 10.1023/A:1007743325440
author: Liangqing Li
published in: \(K\)-Theory
publication date: 2000-09-21T00:00:00Z
review text: For an algebra \(A\) of the so-called class AH, (i.e. which can be presented as the inductive limit of an inductive system \[ P_1M_n({\mathbb C}(X_1) \to P_2M_n({\mathbb C}(X_2) \to \dots , \] where \(X_i\) are finite CW-complexes such that \(\sup\dim X_i < \infty\), \(P_i \in M_{n_i}({\mathbb C}(X_i))\) are projections and for a finite CW-complex \(X\), which is a disjoint union \(X_1 \cup X_2 \cup \dots \cup X_n\) of connected CW-complexes,) the author proves (Theorem 2.3) that if \(A \neq M_n({\mathbb C})\) is a simple AH algebra and \(A \neq {\mathcal K}\), an element \(\alpha \in KK({\mathbb C}(X),A)\) can be presented as \(\alpha=KK(\Phi)\) for some homomorphism \(\Phi:{\mathbb C} \to A\) if and only if \(\alpha\) belongs to the class \(KK_D({\mathbb C}(X),A)\), i.e. admits two properties: (1) The image \(\alpha([1])\) of the unit \(1\in {\mathbb C}(X)\), as an element in \(K_0(A)\) can be presented by an projection \(p=p^*=p^2\). (2) Each component \(\alpha_i \in KK({\mathbb C}(X_i),A)\) in the decomposition \(\alpha = \alpha_1+ \dots + \alpha_n\) either is \(0\) or maps \(K_0({\mathbb C}(X_i))^+ \setminus (0)\) into \(K_0(A)^+\setminus (0)\), the strictly positive property. As the main consequence, the author proves a characterization theorem (Theorem 3.3): If the algebra \(A\) admits a unique trace \(\tau\), then for every element \(\alpha\) with the two properties in the main result above, and for any strictly positive probability measure \(\lambda\in\mathcal (X)\), there is an (injective) homomorphism \(\Phi : {\mathbb C}(X) \to A\) such that: \(KK(\Phi) = \alpha\) and \(m_\Phi := \tau\circ \Phi = \lambda\).
reviewed by: Do Ngoc Diep
Mathematics Subject Classification ID: 46L80
Mathematics Subject Classification ID: 19K35
Mathematics Subject Classification ID: 19K14
zbMATH DE Number: 1384365
zbMATH Keywords: KK-theory
zbMATH Keywords: AF-algebras
zbMATH Keywords: inductive limit
zbMATH Keywords: inductive system
zbMATH Keywords: CW-complexes
zbMATH Keywords: AH algebra
zbMATH Keywords: characterization theorem
MaRDI profile type: Publication
full work available at URL: a:1007743325440
OpenAlex ID: W1998585427
Recommended article: Q5186184
Recommended article: Homomorphisms of \(\mathrm{C}^\ast\)-algebras and their \(\mathrm{K}\)-theory
Recommended article: $ K$-HOMOLOGY OF $ C^*$-ALGEBRAS
Recommended article: Q2943326
Recommended article: Q3441469
Recommended article: Relative \(K\) homology and \(C^*\) algebras
Recommended article: Q3341417
Recommended article: Q3717626
Recommended article: The KK-theory of fundamental C*-algebras
Recommended article: Q3223545
description: scientific article; zbMATH DE number 1384365