Analytic subalgebras of Beurling-Fourier algebras and complexification of Lie groups
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914644907524096 |
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| author | Lee, Heon Lee, Hun Hee |
| author_facet | Lee, Heon Lee, Hun Hee |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_03763 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Analytic subalgebras of Beurling-Fourier algebras and complexification of Lie groups Lee, Heon Lee, Hun Hee Functional Analysis Operator Algebras 46J15, 43A30, 43A40 (Primary), 43A80, 22E25 (Secondary) 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. |
| title | Analytic subalgebras of Beurling-Fourier algebras and complexification of Lie groups |
| topic | Functional Analysis Operator Algebras 46J15, 43A30, 43A40 (Primary), 43A80, 22E25 (Secondary) |
| url | https://arxiv.org/abs/2401.03763 |