The ranges of \(K\)-theoretic invariants for nonsimple graph algebras
type: Item
instance of: scholarly article
zbMATH Open document ID: 1350.46042
published in: Transactions of the American Mathematical Society
publication date: 2016-03-07T00:00:00Z
full work available at URL: https://arxiv.org/abs/1202.1989
Mathematics Subject Classification ID: 46L35
Mathematics Subject Classification ID: 46L80
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 46L45
Mathematics Subject Classification ID: 46L85
zbMATH DE Number: 6551079
zbMATH Keywords: \(C^*\)-algebra
zbMATH Keywords: \(K\)-theory
zbMATH Keywords: six-term exact sequence
zbMATH Keywords: classification
zbMATH Keywords: range of invariant
zbMATH Keywords: extension
MaRDI profile type: Publication
OpenAlex ID: W3105742085
arXiv ID: 1202.1989
cites work: Nonstable \(K\)-theory for graph algebras.
cites work: The K-theoretical range of Cuntz-Krieger algebras
cites work: Reduction of filtered K-theory and a characterization of Cuntz-Krieger algebras
cites work: Corners of Cuntz-Krieger algebras
cites work: The ideal structure of the \(C^*\)-algebras of infinite graphs
cites work: Q4222027
cites work: \(C^*\)-algebras of real rank zero
cites work: Q3670005
cites work: Index maps in the \(K\)-theory of graph algebras
cites work: Stabe rank of graph algebras: Type I graph algebras and their limits
cites work: Viewing AF-algebras as graph algebras
cites work: The \(C^*\)-algebras of arbitrary graphs
cites work: Computing \(K\)-theory and \(\operatorname{Ext}\) for graph \(C^*\)-algebras
cites work: Dimension Groups and Their Affine Representations
cites work: Q3424904
cites work: Classifying \(C^\ast\)-algebras with both finite and infinite subquotients
cites work: The Ordered K-theory of a Full Extension
cites work: Classification of extensions of classifiable \(C^{*}\)-algebras
cites work: On the classification of nonsimple graph \(C^{*}\)-algebras
cites work: On the classification of inductive limits of sequences of semisimple finite-dimensional algebras
cites work: Q3951085
cites work: Extensions for AF C ∗ Algebras and Dimension Groups
cites work: Flow equivalence of reducible shifts of finite type and Cuntz-Krieger algebras.
cites work: The classification of two-component Cuntz-Krieger algebras
cites work: Realizations of AF-algebras as graph algebras, Exel-Laca algebras, and ultragraph algebras
cites work: Q4866042
cites work: Non-simple purely infinite C*-algebras
cites work: $S$-regularity and the corona factorization property
cites work: The simplicity of the quotient algebra $M(A)/A$ of a simple $C^*$-algebra.
cites work: Homology
cites work: C\(^*\)-algebras over topological spaces: filtrated \(K\)-theory
cites work: Q3414354
cites work: A classification theorem for nuclear purely infinite simple \(C^*\)-algebras
cites work: Cuntz-Krieger algebras of infinite graphs and matrices
cites work: Classification of Cuntz-Krieger algebras up to stable isomorphism
cites work: Classification of extensions of certain \(C^*\)-algebras by their six term exact sequences in \(K\)-theory
cites work: Q3149667
cites work: Q4520088
cites work: The range of K-Invariants for C*-Algebras of infinite graphs
cites work: Q4461894
cites work: A Property of Purely Infinite Simple C ∗ -Algebras
DOI: 10.1090/TRAN/6443
title: The ranges of \(K\)-theoretic invariants for nonsimple graph algebras
author: Søren Eilers
author: Takeshi Katsura
author: Mark Tomforde
author: James E. West
review text: The authors obtain the following main results, among many others (cf. [\textit{S. Eilers} and \textit{M. Tomforde}, Math. Ann. 346, No. 2, 393--418 (2010; Zbl 1209.46035)], [\textit{S. Eilers} et al., J. Funct. Anal. 265, No. 3, 449--468 (2013; Zbl 1294.46052); corrigendum ibid. 270, No. 2, 854--859 (2016; Zbl 1355.46051)], [\textit{S. Eilers} and \textit{G. Restorff}, in: Operator algebras. The Abel symposium 2004. Proceedings of the first Abel symposium, Oslo, Norway, September 3--5, 2004. Berlin: Springer. 87--96 (2006; Zbl 1118.46053)] and more).NEWLINENEWLINETheorem 6.1. For the class of graph \(C^*\)-algebras, each of which is an extension of an AF \(C^*\)-algebra that is the largest non-trivial ideal by a purely infinite \(C^*\)-algebra, the six-term exact sequence of \(K\)-theory groups associated to such an extension, with pre-ordered abelian groups, is a complete stable isomorphism invariant, and the range of this invariant consists of all six-term exact sequences of countable abelian groups at six positions from the trivial group to a free abelian group, to another free abelian group, to a Riesz group, to a group with trivial pre-ordering, to another group with trivial pre-ordering, and back to zero. Similarly,NEWLINENEWLINETheorem 6.2. In Theorem 6.1, replacing the largest ideal with the smallest, a similar assertion holds with the range of six-term diagrams from zero to group with the trivial per-ordering, to another with trivial or non-trivial, to a Riesz group, to a free abelian group, to another free abelian group, and back to zero.NEWLINENEWLINETheorem 6.3. In Theorem 6.1, assuming that a graph \(C^*\)-algebra has a unique non-trivial ideal, a similar assertion holds with the range of six-term diagrams in the four cases where the ideal and the quotient are either both Kirchberg \(C^*\)-algebras, or an AF \(C^*\)-algebra and a Kirchberg algebra, or a Kirchberg algebra and an AF, or both AF, respectively. Moreover,NEWLINENEWLINETheorem 6.4. For the cl…
reviewed by: Takahiro Sudo
description: scientific article; zbMATH DE number 6551079