Algebras whose groups of units are Lie groups

Volume 153 / 2002

Helge Glöckner Studia Mathematica 153 (2002), 147-177 MSC: 22E65, 46E25, 46F05, 46H05, 46H30. DOI: 10.4064/sm153-2-4


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.


  • Helge GlöcknerFB Mathematik
    TU Darmstadt
    Schlossgartenstr. 7
    64289 Darmstadt, Germany

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