Old and new results on Allan's $GB^*$-algebras
Volume 91 / 2010
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.
