Algebras whose groups of units are Lie groups. (English) Zbl 1009.22021

Glöckner, Helge

Stud. Math. 153, No. 2, 147-177 (2002).

The 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

Keywords:

group of units; continuous inverse algebra; invertible elements; root-graded Lie groups; functional calculus