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 𝓐₀[||·||₀], under a locally convex *-algebra topology τ, making the multiplication of 𝓐₀ jointly continuous. We conclude that such a completion is a GB*-algebra over the τ-closure of the unit ball of 𝓐₀[||·||₀]. Further, we discuss some consequences of this result; we briefly comment the case when τ makes the multiplication of 𝓐₀ separately continuous and illustrate the results by examples.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-bc91-0-9,
author = {Maria Fragoulopoulou and Atsushi Inoue and Klaus-Detlef K\"ursten},
title = {Old and new results on Allan's GB*-algebras},
journal = {Banach Center Publications},
volume = {89},
year = {2010},
pages = {169-178},
zbl = {1213.46041},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc91-0-9}
}
Maria Fragoulopoulou; Atsushi Inoue; Klaus-Detlef Kürsten. Old and new results on Allan's GB*-algebras. Banach Center Publications, Tome 89 (2010) pp. 169-178. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc91-0-9/