Geometry and topology of operators on Hilbert \(C^*\)-modules
type: Item
instance of: scholarly article
title: Geometry and topology of operators on Hilbert \(C^*\)-modules
zbMATH Open document ID: 0959.46043
DOI: 10.1007/BF02355449
published in: Journal of Mathematical Sciences (New York)
publication date: 2000-08-27T00:00:00Z
review text: The paper surveys the theory of operators on Hilbert \(C^*\)-modules. The author starts with multiplier \(C^*\)-algebras and discusses various topologies on the set of operators on a Hilbert \(C^*\)-module \(M\) over a \(C^*\)-algebra \(A\) and relations between various classes of operators on \(M\) and multipliers of the \(C^*\)-algebra of compact operators on \(M\). A number of examples is given and the commutative case is discussed in detail. The main interest of the author lies in topological properties of the full linear group of the algebra of bounded/adjointable operators on the standard Hilbert \(C^*\)-module \(\ell_2(A)\). A variant of the Dixmier-Douady theorem is proved, showing that every bounded set of invertible operators is contractible in the full linear group with respect to the left strict topology. The author gives a new short proof of the Cuntz--Higson theorem on the contractibility of the unitary group of operators on \(\ell_2(A)\) with respect to the norm topology and gives some new partial results on the contractibility of bounded sets of invertible bounded operators (non necessarily adjointable) with respect to the norm topology.
Mathematics Subject Classification ID: 46L08
zbMATH DE Number: 1495336
zbMATH Keywords: Hilbert \(C^*\)-module
zbMATH Keywords: operator
zbMATH Keywords: strict topology
zbMATH Keywords: multiplier algebra
zbMATH Keywords: contractibility
zbMATH Keywords: \(C^*\)-algebra of compact operators
zbMATH Keywords: topological properties of the full linear group
zbMATH Keywords: Dixmier-Douady theorem
zbMATH Keywords: Cuntz-Higson theorem
author: E. V. Troitskij
reviewed by: Vladimir M. Manuilov
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1495336