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## Old and new results on Allan's $GB^*$-algebras

### Volume 91 / 2010

Banach Center Publications 91 (2010), 169-178 MSC: Primary 46H20; Secondary 47L60. DOI: 10.4064/bc91-0-9

#### Abstract

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.

#### Authors

• Maria FragoulopoulouDepartment of Mathematics
University of Athens
Panepistimiopolis, Athens 15784, Greece
e-mail
• Atsushi InoueDepartment of Applied Mathematics
Fukuoka University
Nanakuma, Jonan-ku
Fukuoka 814-0180, Japan
e-mail
• Klaus-Detlef KürstenMathematisches Institut
Universität Leipzig
Johannisgasse 26
D-04103 Leipzig, Germany
e-mail

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