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arXiv · 2401.03763

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

Heon Lee·Hun Hee Lee
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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.

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BibTeXRIS

Heon Lee, Hun Hee Lee. 2024-01-08. Analytic subalgebras of Beurling-Fourier algebras and complexification of Lie groups. https://arxiv.org/abs/2401.03763

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