Trace-preserving homomorphisms of semigroups.
type: Item
instance of: scholarly article
title: Trace-preserving homomorphisms of semigroups.
zbMATH Open document ID: 1051.47030
DOI: 10.1016/S0022-1236(03)00072-7
author: Milan Hladnik
author: M. Omladič
author: H. Radjavi
published in: Journal of Functional Analysis
publication date: 2004-02-03T00:00:00Z
review text: The authors study the structure of trace-preserving homomorphisms of semigroups of trace-class operators. Let \({\mathcal S}\) be a semigroup of trace-class operators acting on a complex Hilbert space \(H\). If the closed subalgebra generated by \({\mathcal S}\) in the Banach algebra \({\mathcal C}_1\) of all trace-class operators is semisimple, then the semigroup \({\mathcal S}\) is called semisimple. The main result of the paper reads as follows. Let \(\phi: {\mathcal S} \to {\mathcal T}\) be a surjective trace-preserving homomorphism between semisimple semigroups of trace-class operators. If \({\mathcal S}\) and \({\mathcal T}\) satisfy certain totality conditions, then \(\phi\) is induced by a closed, injective operator \(V\) with dense domain and dense range in the following sense. For every \(S\in {\mathcal S}\), we have \(\phi(S)V=VS\) on the domain of \(V\). Some discussions and examples are given to understand the assumptions in this and in some preliminary results. Several applications of the main result in the finite and infinite dimensions are also presented. As for finite dimension, it is shown that every irreducible semigroup of operators individually similar to \(J\)-unitaries is similar to a semigroup of \(J\)-unitaries with one and the same \(J\). An analogous result holds for unitaries. As for infinite dimension, it is proved that if the elements of an irreducible semigroup of operators of the form \(F+\lambda I\), \(F\) being of finite rank, are individually similar to unitaries, then the whole semigroup is similar to a semigroup of unitaries via some densely defined, injective, positive operator with dense range.
reviewed by: Lajos Molnár
Mathematics Subject Classification ID: 47B49
zbMATH DE Number: 2037115
zbMATH Keywords: trace-preserving homomorphisms
zbMATH Keywords: semigroups of trace-class operators
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2037115