On restricted perturbations in inverse images and a description of normalizer algebras in \(C^*\)-algebras
type: Item
instance of: scholarly article
title: On restricted perturbations in inverse images and a description of normalizer algebras in \(C^*\)-algebras
zbMATH Open document ID: 0828.46052
author: Eberhard Kirchberg
published in: Journal of Functional Analysis
publication date: 1996-01-11T00:00:00Z
review text: Let \(E\) and \(F\) be Banach spaces and \(T\) a bounded linear map from \(E\) into \(F\). Using a certain perturbation function \(f(\cdot, T)\) the author finds a useful sufficient criterion that \(T\) maps the closed unit ball onto a closed set. Applying this to the quotient map from a \(C^*\)- algebra \(A\) onto its quotient Banach space \(A/ (L+ R)\) by the sum \(L+R\) of closed left and right ideals \(L\) and \(R\) of \(A\) he obtains that the closed unit ball of \(A\) maps onto the closed unit ball of \(A/ (L+ R)\). This gives an independent description of the images of the right normalizer algebra \(N_r (D)\) and the normalizer algebra \(N(D)= N_l (D)\cap N_r (D)\) of a hereditary \(C^*\)-subalgebra \(D\) of \(A\) by the quotient map from \(A\) onto \(A/(\text{cl} (AD)+ \text{cl} (DA))\). Then he uses this to prove a necessary and sufficient criterion for a \(C^*\)- algebra \(B\) to be a \(C^*\)-quotient-algebra of a \(C^*\)-subalgebra of a \(C^*\)-algebra \(A\). The latter criterion will be applied in the situation where \(A\) equals the CAR-algebra in some forthcoming papers.
Mathematics Subject Classification ID: 46L05
zbMATH DE Number: 764016
zbMATH Keywords: restricted perturbations
zbMATH Keywords: normalizer algebras in \(C^*\)-algebras
zbMATH Keywords: inverse images
zbMATH Keywords: perturbation of unitaries
zbMATH Keywords: \(C^*\)-spaces
zbMATH Keywords: \(C^*\)- systems
zbMATH Keywords: normalizer algebras
zbMATH Keywords: perturbation function
zbMATH Keywords: \(C^*\)-quotient- algebra
zbMATH Keywords: \(C^*\)-subalgebra
zbMATH Keywords: CAR-algebra
MaRDI profile type: Publication
full work available at URL: jfan.1995.1040
OpenAlex ID: W2053606045
DOI: 10.1006/JFAN.1995.1040
description: scientific article; zbMATH DE number 764016