Completeness of Infinite-dimensional Lie Groups in Their Left Uniformity

Glöckner, Helge

Canadian journal of mathematics, Tome 71 (2019) no. 1, pp. 131-152

Résumé

We prove completeness for the main examples of infinite-dimensional Lie groups and some related topological groups. Consider a sequence $G_{1}\subseteq G_{2}\subseteq \cdots \,$ of topological groups $G_{n}$ n such that $G_{n}$ is a subgroup of $G_{n+1}$ and the latter induces the given topology on $G_{n}$, for each $n\in \mathbb{N}$. Let $G$ be the direct limit of the sequence in the category of topological groups. We show that $G$ induces the given topology on each $G_{n}$ whenever $\cup _{n\in \mathbb{N}}V_{1}V_{2}\cdots V_{n}$ is an identity neighbourhood in $G$ for all identity neighbourhoods $V_{n}\subseteq G_{n}$. If, moreover, each $G_{n}$ is complete, then $G$ is complete. We also show that the weak direct product $\oplus _{j\in J}G_{j}$ is complete for each family $(G_{j})_{j\in J}$ of complete Lie groups $G_{j}$. As a consequence, every strict direct limit $G=\cup _{n\in \mathbb{N}}G_{n}$ of finite-dimensional Lie groups is complete, as well as the diffeomorphism group $\text{Diff}_{c}(M)$ of a paracompact finite-dimensional smooth manifold $M$ and the test function group $C_{c}^{k}(M,H)$, for each $k\in \mathbb{N}_{0}\cup \{\infty \}$ and complete Lie group $H$ modelled on a complete locally convex space.

Détail

DOI : 10.4153/CJM-2017-048-5

Mots-clés : infinite-dimensional Lie group, direct sum, box product, weak direct product, (LB)-space, inductive limit, direct limit, ascending sequence, product set, bamboo shoot topology, compact support, test function group, diffeomorphism group, Banach–Lie group, left uniform structure, Cauchy net, Cauchy filter, strong (ILB)-Lie group, projective limit, inverse limit