Algebras whose groups of units are Lie groups
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.