Algebras whose groups of units are Lie groups

Helge Glöckner

Helge Glöckner

Studia Mathematica, Tome 151 (2002), p. 147-177 / Harvested from The Polish Digital Mathematics Library

Résumé

Let A be a locally convex, unital topological algebra whose group of units $A^{\times}$ is open and such that inversion $ι : A^{\times} \to A^{\times}$ is continuous. Then inversion is analytic, and thus $A^{\times}$ is an analytic Lie group. We show that if A is sequentially complete (or, more generally, Mackey complete), then $A^{\times}$ has a locally diffeomorphic exponential function and multiplication is given locally by the Baker-Campbell-Hausdorff series. In contrast, for suitable non-Mackey complete A, the unit group $A^{\times}$ is an analytic Lie group without a globally defined exponential function. We also discuss generalizations in the setting of “convenient differential calculus”, and describe various examples.

Publié le : 2002-01-01

Zbl 1009.22021

EUDML-ID : urn:eudml:doc:284755

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Helge Glöckner. Algebras whose groups of units are Lie groups. Studia Mathematica, Tome 151 (2002) pp. 147-177. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm153-2-4/