Analytic subalgebras of Beurling-Fourier algebras and complexification of Lie groups

TL;DR

This paper explores the structure of analytic subalgebras within Beurling-Fourier algebras on connected Lie groups, demonstrating local and global solutions for complexification evaluations, especially in nilpotent and specific non-nilpotent cases.

Contribution

It introduces new analytic subalgebras enabling local and global evaluations on complexified Lie groups, extending understanding of Fourier algebra structures.

Findings

Local solutions for general connected Lie groups

Global solutions for simply connected, nilpotent Lie groups

Analysis of the $ax+b$-group case

Abstract

In this paper, we focus on how we can interpret the actions of the elements in the Gelfand spectrum of a weighted Fourier algebra on connected Lie groups. They can be viewed as evaluations on specific points of the complexification of the underlying Lie group by restricting to a particular dense subalgebra, which we call an analytic subalgebra. We first introduce an analytic subalgebra allowing a ``local" solution for general connected Lie groups. We will demonstrate that a ``global" solution is also possible for connected, simply connected and nilpotent Lie groups through a different choice of an analytic subalgebra. Finally, we examine the case of the $ax+b$-group as an example of a non-nilpotent, non-unimodular Lie group with a ``global" solution.