Algebras whose groups of units are Lie groups.

*(English)*Zbl 1009.22021The continuous inverse algebras (commutative or not) are investigated with emphasis on applications in infinite-dimensional Lie group theory. One considers a locally convex, unital topological algebra \(A\) whose group of units \(A^{\times }\) is open and such that the inverse map \(i:A^{\times }\rightarrow A^{\times }\) is continuous. It is shown that the inversion map is also analytic and that the group of units \(A^{\times }\) is an analytic Lie group. Supposing that \(A\) is sequential complete (or, more generally, Mackey complete), it is proven that \(A^{\times }\) is a Baker-Campbell-Hausdorff (BCH)-Lie group, i.e., an analytic Lie group with a locally diffeomorphic exponential function whose multiplication is given locally by the BCH-series. In contrast, for suitable non-Mackey complete algebras \(A\), it is shown that the corresponding group of units \(A^{\times }\) is also an analytic group but without a globally defined exponential function. It is argued that the Mackey complete continuous inverse algebras provide a particularly well suited class of coordinate domains for root-graded Lie groups. Some generalizations in the setting of ”convenient differential calculus” are discussed and various examples of continuous inverse algebras are presented.

Reviewer: Gheorghe Zet (Iasi)

##### MSC:

22E65 | Infinite-dimensional Lie groups and their Lie algebras: general properties |

46E25 | Rings and algebras of continuous, differentiable or analytic functions |

46F05 | Topological linear spaces of test functions, distributions and ultradistributions |

46H05 | General theory of topological algebras |

46H30 | Functional calculus in topological algebras |