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

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
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