Trivial \(K_ 1\)-flow of AF algebras and finite von Neumann algebras
type: Item
instance of: scholarly article
title: Trivial \(K_ 1\)-flow of AF algebras and finite von Neumann algebras
zbMATH Open document ID: 0673.46039
DOI: 10.1016/0022-1236(90)90068-V
author: Shuang Zhang
published in: Journal of Functional Analysis
publication date: 1990-01-01T00:00:00Z
review text: A \(C^*\)-algebra \({\mathcal A}\) is said to have a trivial \(K_ 1\)-flow if \(K_ 1({\mathcal B})=0\) for any hereditary \(C^*\)-subalgebra \({\mathcal B}\) of M(\({\mathcal A})\). We prove that if \({\mathcal A}\) is a \(\sigma\)-unital AF algebra, then \({\mathcal A}\) has a trivial \(K_ 1\)-flow, and that if \({\mathcal A}\) is a finite von Neumann algebra, then \({\mathcal A}\otimes {\mathcal K}\) has a trivial \(K_ 1\)-flow. If the multiplier algebra of \({\mathcal A}\) has the FS property, then the unitary group of \(\tilde {\mathcal B}\) is connected for any hereditary \(C^*\)-subalgebra \({\mathcal B}\) of M(\({\mathcal A})\). We also prove that if \({\mathcal A}\) is either a \(\sigma\)-unital nonunital purely infinite simple \(C^*\)-algebra or a nonunital \(C^*\)-algebra stably isomorphic to a Bunce-Deddens algebra, then \(K_ 1({\mathcal B})=0\) for any hereditary \(C^*\)-subalgebra \({\mathcal B}\) of M(\({\mathcal A})\) not contained in \({\mathcal A}\).
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 46L80
zbMATH DE Number: 4101890
zbMATH Keywords: trivial \(K_ 1\)-flow
zbMATH Keywords: hereditary \(C^*\)-subalgebra
zbMATH Keywords: sigma-unital AF algebra
zbMATH Keywords: finite von Neumann algebra
zbMATH Keywords: multiplier algebra
zbMATH Keywords: FS property
zbMATH Keywords: unitary group
zbMATH Keywords: Bunce-Deddens algebra
MaRDI profile type: Publication
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full work available at URL: 0022-1236(90)90068-v
OpenAlex ID: W2043739577
description: scientific article; zbMATH DE number 4101890