Algebras whose groups of units are Lie groups

Helge Glöckner¹

¹ FB Mathematik TU Darmstadt Schlossgartenstr. 7 64289 Darmstadt, Germany

Studia Mathematica, Tome 153 (2002) no. 2, pp. 147-177

Résumé

Let $A$ be a locally convex, unital topological algebra whose group of units $A^\times $ is open and such that inversion $\iota : 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.

Détail

DOI : 10.4064/sm153-2-4

Keywords: locally convex unital topological algebra whose group units times inversion iota times times continuous inversion analytic times analytic lie group sequentially complete generally mackey complete times has locally diffeomorphic exponential function multiplication given locally baker campbell hausdorff series contrast suitable non mackey complete unit group times analytic lie group without globally defined exponential function discuss generalizations setting convenient differential calculus describe various examples