The Regularity Problem for Lie Groups with Asymptotic Estimate Lie Algebras

TL;DR

This paper addresses the regularity problem for infinite-dimensional Lie groups with asymptotic estimate Lie algebras, establishing conditions under which the evolution map is smooth and characterizing regularity in terms of completeness and convexity properties.

Contribution

It provides a characterization of the regularity of Milnor's infinite-dimensional Lie groups with asymptotic estimate Lie algebras, linking smoothness of the evolution map to topological and algebraic conditions.

Findings

The evolution map is $C^ abla$-continuous iff it is $C^0$-continuous on certain domains.

$G$ is $k$-confined if constricted, a condition close to asymptotic estimate.

Asymptotic estimate Lie groups are $C^ abla$-regular iff they are Mackey complete, locally $ ext{μ}$-convex, and have Mackey complete Lie algebra.

Abstract

We solve the regularity problem for Milnor's infinite dimensional Lie groups in the asymptotic estimate context. Specifically, let $G$ be a Lie group with asymptotic estimate Lie algebra $\mathfrak{g}$, and denote its evolution map by $\mathrm{evol}\colon \mathrm{D}\equiv \mathrm{dom}[\mathrm{evol}]\rightarrow G$, i.e., $\mathrm{D}\subseteq C^0([0,1],\mathfrak{g})$. We show that $\mathrm{evol}$ is $C^\infty$-continuous on $\mathrm{D}\cap C^\infty([0,1],\mathfrak{g})$ if and only if $\mathrm{evol}$ is $C^0$-continuous on $\mathrm{D}\cap C^0([0,1],\mathfrak{g})$. We furthermore show that $G$ is k-confined for $k\in \mathbb{N}\sqcup\{\mathrm{lip},\infty\}$ if $G$ is constricted. (The latter condition is slightly less restrictive than to be asymptotic estimate.) Results obtained in a previous paper then imply that an asymptotic estimate Lie group $G$ is $C^\infty$-regular if and only if it is Mackey complete, locally $\mu$-convex, and has Mackey complete Lie algebra - In this case, $G$ is $C^k$-regular for each $k\in \mathbb{N}_{\geq 1}\sqcup\{\mathrm{lip},\infty\}$ (with ``smoothness restrictions'' for $k\equiv\mathrm{lip}$), as well as $C^0$-regular if $G$ is even sequentially complete with integral complete Lie algebra.