On the range of the Elliott invariant
type: Item
instance of: scholarly article
title: On the range of the Elliott invariant
zbMATH Open document ID: 0816.46075
author: Klaus Thomsen
published in: Journal of Functional Analysis
publication date: 1995-07-16T00:00:00Z
review text: Let \(A\) be a unital \(C^*\)-algebra. For any tracial state \(\omega\) on \(A\) there is a natural way to define a state \(r_ A(\omega)\) of the \(K_ 0\)-group of \(A\). \(r_ A\) is an affine continuous map from the tracial state space of \(A\) to that of \(K_ 0(A)\). This map enters as a crucial ingredient in the invariant used by Elliott to classify the simple unital \(C^*\)-algebras that arise as inductive limits of sequences of finite direct sums of matrix algebras over \(C[0,1]\). The author shows that for such \(C^*\)-algebras (and many others) the map \(r_ A\) must preserve extreme points, and that any continuous affine surjection between metrizable Choquet simplices which preserves extreme points and is open can be realized as the \(r_ A\)-map corresponding to a simple unital inductive limit \(C^*\)-algebra of a sequence of finite direct sums of matrix algebras over \(C[0,1]\).
Mathematics Subject Classification ID: 46L80
zbMATH DE Number: 599783
zbMATH Keywords: unital \(C^*\)-algebra
zbMATH Keywords: \(K_ 0\)-group
zbMATH Keywords: affine continuous map
zbMATH Keywords: tracial state space
zbMATH Keywords: inductive limits of sequences of finite direct sums of matrix algebras over \(C[0,1]\)
zbMATH Keywords: continuous affine surjection between metrizable Choquet simplices which preserves extreme points and is open
zbMATH Keywords: simple unital inductive limit \(C^*\)-algebra
MaRDI profile type: Publication
full work available at URL: jfan.1994.1049
OpenAlex ID: W2062824923
DOI: 10.1006/JFAN.1994.1049
description: scientific article; zbMATH DE number 599783