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Old and new results on Allan’s \(GB^*\)-algebras. (English) Zbl 1213.46041

Loy, Richard J. (ed.) et al., Banach algebras 2009. Proceedings of the 19th international conference, Bȩdlewo, Poland, July 14–24, 2009. Warszawa: Polish Academy of Sciences, Institute of Mathematics (ISBN 978-83-86806-10-2/pbk). Banach Center Publications 91, 169-178 (2010).
Summary: This is an expository paper on the importance and applications of \(GB^*\)-algebras in the theory of unbounded operators, which is closely related to quantum field theory and quantum mechanics. After recalling the definition and the main examples of \(GB^*\)-algebras, we exhibit their most important properties. Then, through concrete examples we are led to a question concerning the structure of the completion of a given \(C^*\)-algebra \(\mathcal A_0[\|\cdot\|_0]\), under a locally convex \(*\)-algebra topology \(\tau\), making the multiplication of \(\mathcal A_0\) jointly continuous. We conclude that such a completion is a \(GB^*\)-algebra over the \(\tau\)-closure of the unit ball of \(\mathcal A_0[\|\cdot\|_0]\). Further, we discuss some consequences of this result; we briefly comment the case when \(\tau\) makes the multiplication of \(\mathcal A_0\) separately continuous and illustrate the results by examples.
For the entire collection see [Zbl 1206.46004].

MSC:

46H20 Structure, classification of topological algebras
47L60 Algebras of unbounded operators; partial algebras of operators
46K99 Topological (rings and) algebras with an involution
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