Analytic Mappings Between LB-spaces and Applications in Infinite-Dimensional Lie Theory

TL;DR

This paper establishes a criterion for complex analyticity of nonlinear maps on direct limits of normed spaces and applies it to construct new classes of infinite-dimensional Lie groups, advancing the understanding of infinite-dimensional Lie theory.

Contribution

It introduces a sufficient criterion for complex analyticity on direct limits of normed spaces and uses it to build novel infinite-dimensional Lie groups.

Findings

Developed a criterion for complex analyticity of nonlinear maps.

Constructed new classes of infinite-dimensional Lie groups.

Applied the criterion to groups of germs and ascending unions of Banach Lie groups.

Abstract

We give a sufficient criterion for complex analyticity of nonlinear maps defined on direct limits of normed spaces. This tool is then used to construct new classes of (real and complex) infinite dimensional Lie groups: (a) groups of germs of analytic diffeomorphisms around a compact set in a Banach space and (b) unions of ascending sequences of Banach Lie groups.