Nonstandard Hulls of Locally Exponential Lie Algebras

TL;DR

This paper develops a method to construct nonstandard hulls of infinite-dimensional Lie algebras, extending Pestov's theorem, and introduces a nonstandard smoothness condition to ensure smoothness of induced functions.

Contribution

It generalizes Pestov's enlargeability theorem for Banach-Lie algebras to certain infinite-dimensional cases using nonstandard analysis techniques.

Findings

Constructed nonstandard hulls for specific infinite-dimensional Lie algebras.

Established a nonstandard smoothness condition for functions between locally convex spaces.

Provided conditions ensuring nonstandard extensions map finite points to finite points.

Abstract

We show how to construct the nonstandard hull of certain infinite-dimensional Lie algebras in order to generalize a theorem of Pestov on the enlargeability of Banach-Lie algebras. In the process, we consider a nonstandard smoothness condition on functions between locally convex spaces to ensure that the induced function between the nonstandard hulls is smooth. We also discuss some conditions on a function between locally convex spaces which guarantee that its nonstandard extension maps finite points to finite points.