Classifying \(C^*\)-algebras via ordered, mod-\(p\) \(K\)-theory
type: Item
instance of: scholarly article
title: Classifying \(C^*\)-algebras via ordered, mod-\(p\) \(K\)-theory
zbMATH Open document ID: 0857.46039
DOI: 10.1007/BF01444239
author: Terry A. Loring
author: Marius Dadarlat
published in: Mathematische Annalen
publication date: 1997-03-10T00:00:00Z
full work available at URL: https://eudml.org/doc/183005
review text: We introduce an order structure on \(K_0(-)\oplus K_0(-; \mathbb{Z}/p)\). This group may also be thought of as \(K_0(-; \mathbb{Z}\oplus \mathbb{Z}/p)\). We exhibit new examples of real-rank zero \(C^*\)-algebras that are inductive limits of finite-dimensional and dimension-drop algebras, have the same ordered, graded \(K\)-theory with order unit and yet are not isomorphic. In fact they are not even stably shape equivalent. The order structure on \(K_0(-; \mathbb{Z}\oplus\mathbb{Z}/p)\) naturally distinguishes these algebras. The same invariant is used to give an isomorphism theorem for such real-rank zero inductive limits. As a corollary we obtain an isomorphism theorem for all real-rank zero approximately homogeneous \(C^*\)-algebras that arise from systems of bounded dimension growth and torsion-free \(K_0\) group.
Mathematics Subject Classification ID: 46L35
Mathematics Subject Classification ID: 46L80
zbMATH DE Number: 931998
zbMATH Keywords: order structure
zbMATH Keywords: real-rank zero \(C^*\)-algebras
zbMATH Keywords: graded \(K\)-theory
zbMATH Keywords: real-rank zero inductive limits
zbMATH Keywords: real-rank zero approximately homogeneous \(C^*\)-algebras
zbMATH Keywords: torsion-free \(K_ 0\) group
MaRDI profile type: Publication
cites work: Q3730469
cites work: Shape theory for $C^*$-algebras.
cites work: Symmetries of the CAR algebra
cites work: Inductive Limits of Finite Dimensional C ∗ -Algebras
cites work: Finite group actions on AF algebras obtained by folding the interval
cites work: \(C^*\)-algebras of real rank zero
cites work: Shape theory and asymptotic morphisms for \(C^*\)-algebras
cites work: Approximately unitarily equivalent morphisms and inductive limit \(C^*\)- algebras
cites work: Reduction to dimension three of local spectra of real rank zero C*-algebras.
cites work: \(K\)-homology, asymptotic representations, and unsuspended \(E\)-theory
cites work: Q3788535
cites work: On the classification of inductive limits of sequences of semisimple finite-dimensional algebras
cites work: On the classification of C*-algebras of real rank zero.
cites work: Compact group actions on UHF algebras obtained by folding the interval
cites work: THE OPERATORK-FUNCTOR AND EXTENSIONS OFC*-ALGEBRAS
cites work: The \(K\)-theory of AF embeddings of the rational rotation algebras
cites work: The Künneth theorem and the universal coefficient theorem for Kasparov's generalized K-functor
cites work: Topological methods for \(C^*\)-algebras. IV: Mod p homology
OpenAlex ID: W1990299992
description: scientific article; zbMATH DE number 931998