The stable rank of some free product \(C^*\)-algebras
type: Item
instance of: scholarly article
title: The stable rank of some free product \(C^*\)-algebras
zbMATH Open document ID: 0905.46036
DOI: 10.1215/S0012-7094-97-09004-9
author: Uffe Haagerup
author: Ken Dykema
author: Mikael Rørdam
published in: Duke Mathematical Journal
publication date: 1998-06-08T00:00:00Z
full work available at URL: https://arxiv.org/abs/funct-an/9608001
review text: A unital \(C^*\)-algebra \(A\) is said to have stable rank 1 if the group of invertible elements in \(A\) is a norm dense subset of \(A\). The authors prove a general result giving sufficient conditions for a \(C^*\)-algebra \((U,\tau)\) to have stable rank 1, where \((U,\tau)\) is the reduced free product of a family \((A_i,\tau_i)\), \(i\in I\), of unital \(C^*\)-algebras \(A_i\) with normalized faithful traces \(\tau_i\). In particular, one obtains that the reduced group \(C^*\)-algebra \(C^*_{\text{red}}(G)\) has stable rank 1 if \(G\) is a discrete group arising as a free product \(G_1* G_2\) where \(| G_1|\geq 2\) and \(| G_2|\geq 3\).
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 46L35
zbMATH DE Number: 1160583
zbMATH Keywords: unital \(C^*\)-algebra
zbMATH Keywords: stable rank 1
zbMATH Keywords: group of invertible elements
zbMATH Keywords: reduced free product
zbMATH Keywords: normalized faithful traces
zbMATH Keywords: reduced group \(C^*\)-algebra
reviewed by: Sh. A. Ayupov
MaRDI profile type: Publication
arXiv ID: funct-an/9608001
cites work: Q5287641
cites work: Free Products of C ∗ -Algebras
cites work: Modules Over Coproducts of Rings
cites work: Dimension functions and traces on C*-algebras
cites work: Q5639812
cites work: Purely Infinite, Simple $C^*$-Algebras Arising from Free Product Constructions, II
cites work: Almost commuting self-adjoint matrices - a short proof of Huaxin Lin's theorem.
cites work: An example of a non nuclear C*-algebra, which has the metric approximation property
cites work: Quasitraces on exact C*-algebras are traces
cites work: Q3316025
cites work: Dimension and Stable Rank in the K -Theory of C* -Algebras
cites work: Q3726814
cites work: Q3803668
cites work: Advances in the theory of unitary rank and regular approximation
cites work: On the stable rank of simple C*-algebras
cites work: Q4727689
cites work: Free Random Variables
OpenAlex ID: W1963735393
description: scientific article