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

Heon Lee, Hun Hee Lee

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.

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

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 -group as an example of a non-nilpotent, non-unimodular Lie group with a ``global" solution.
Paper Structure (17 sections, 27 theorems, 200 equations)

This paper contains 17 sections, 27 theorems, 200 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Locally convex spaces: tensor products, quotients and vector-valued integrals
  4. Pettis integral
  5. Unbounded operators
  6. Lie groups, algebras and representations
  7. Analytic and entire vectors
  8. Fourier algebras and weights on the dual of $G$
  9. Beurling-Fourier algebras
  10. Abstract complexification
  11. Analytic subalgebras and their density
  12. Analytic evaluations and a local solution
  13. A global solution for connected, simply connected and nilpotent groups
  14. A modified analytic subalgebra
  15. A partially holomorphic evaluation
  16. ...and 2 more sections

Key Result

Lemma 2.1

Let $T :\mathcal{D}(T)\subseteq \mathcal{H} \to \mathcal{H}$ be a normal operator. For a vector $v\in \bigcap_{n\ge 1}\mathcal{D}(|T|^n)$ with $\sum_{n\ge 0}\frac{\||T|^nv\|}{n!}<\infty$, we have $v\in \mathcal{D}(e^T)$ and where the series converges absolutely in $\mathcal{H}$.

Theorems & Definitions (67)

  • Lemma 2.1
  • proof
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Definition 2.5
  • Remark 2.6
  • Definition 2.7
  • ...and 57 more