Algebras whose groups of units are Lie groups

Abstract

Abstract. Let A be a locally convex, unital topological algebra whose group of units A × is open and such that inversion ι :Then inversion is analytic, and thus A × is an analytic Lie group. We show that if A is sequentially complete (or, more generally, Mackey complete), then A × 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 × 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.

Finite-dimensional Lie subalgebras of algebras with continuous inversion

“…If, in addition, A is complete, then the same arguments as for Banach algebras lead to a holomorphic functional calculus ([Wae67], [Gl02]). Since completeness is in general not inherited by quotients ([Koe69, §31.6]), it is natural to consider for CIAs the weaker condition of FC-completeness, that is, closedness under holomorphic functional calculus.…”

“…If a ∈ S satisfies r A (a) < 1, then 1 − a is invertible and the Neumann series ∞ n=0 a n converges to (1 − a) −1 ( [Gl02]). Since S is closed, (1 − a) −1 ∈ S, so that S × is a neighborhood of 1 in S, hence open.…”

“…[Mil83], [Bel06], [GN06]). In this context, the unit groups of continuous inverse algebras are the prototypical "linear Lie groups" ( [Gl02]), and it is a natural question whether the notion of "linearity" in this general context determines a larger class of finite-dimensional Lie groups than the Lie groups of matrices.…”

Fundamental Problems in the Theory of Infinite-Dimensional Lie Groups

“…Our counterexamples show that neither projective nor direct limits of finite-dimensional (and hence BCH-) Lie groups need to be locally exponential. For further information on BCH-Lie groups and locally exponential Lie groups, see [3], [4], [12], [19], [29] and [30].…”

General Dirichlet series, arithmetic convolution equations and Laplace transforms

“…It is called complete if the underlying locally convex space is complete. A continuous inverse algebra is a unital, associative complex topological algebra A whose group of units A * is open in A and whose inversion map A * → A, a → a −1 , is continuous (see [2], [7] and [12]). The spectrum of a commutative continuous inverse algebra A is the set A of all unital algebra homomorphisms ξ : A → C. It is known that ξ → ker ξ is a bijection from A onto the set of all maximal (proper) ideals of A (cf.…”