Regularity of unbounded operators in \(C^*\)-modules.
type: Item
instance of: scholarly article
title: Regularity of unbounded operators in \(C^*\)-modules.
zbMATH Open document ID: 1162.46031
author: Stéphane Damaville
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2007-08-10T00:00:00Z
review text: Regular operators between Hilbert \(C^*\)-modules are studied. Sufficient conditions for the regularity of the product and of the sum of such operators are provided. It is shown that, if \(d\) is a closed densely defined operator on a \(C^*\)-module \(E\), with densely defined adjoint \(d^*\), such that \(\text{Im}\, d \subset \text{Dom}\,d\), and there exist compact operators \(a,b,k\) on \(E\) whose images are included in \(\text{Dom}\,d\) and satisfy \(da+bd=1+k\) on \(\text{Dom}\,d\), then the operators \(d\) and \(d+d^*\) are regular, and the resolvent \((d+d^*\pm i)^{-1}\) is compact. As an application, the signature operator on a Lip\-schitz manifold endowed with a proper action of a discrete group is shown to define a \(K\)-theory element on the reduced group \(C^*\)-algebra.
reviewed by: Florin P. Boca
Mathematics Subject Classification ID: 46L08
Mathematics Subject Classification ID: 46L80
Mathematics Subject Classification ID: 58B34
zbMATH DE Number: 5178519
zbMATH Keywords: \(C^*\)-modules
zbMATH Keywords: regular unbounded operators
zbMATH Keywords: Lipschitz manifold
zbMATH Keywords: signature operator
MaRDI profile type: Publication
full work available at URL: j.crma.2007.03.020
OpenAlex ID: W2078447711
cites work: Q3344718
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DOI: 10.1016/J.CRMA.2007.03.020
description: scientific article; zbMATH DE number 5178519