# Wydawnictwa / Czasopisma IMPAN / Studia Mathematica / Wszystkie zeszyty

## Algebras whose groups of units are Lie groups

### Tom 153 / 2002

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

#### Streszczenie

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.

#### Autorzy

• Helge GlöcknerFB Mathematik