Arens Products and Asymptotic Structures on Chébli-Trimèche Hypergroups under Low Regularity Conditions
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866908704786350080 |
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| author | Sababe, Saeed Hashemi |
| author_facet | Sababe, Saeed Hashemi |
| contents | We investigate the Arens products on the second duals of convolution algebras associated with Chébli--Trimèche hypergroups, particularly focusing on the left and right topological centres of $L^{1}(H)^{\prime\prime}$ and $M(H)^{\prime\prime}$. Building on the recent framework established by Losert, we relax the classical smoothness assumptions on the underlying Sturm--Liouville function $A$ and develop new asymptotic analysis tools for measure-valued and low-regularity perturbations. This allows us to extend the existence and continuity of the asymptotic measures $ν_{x}$ and the limit measure $ν_{\infty}$ to a strictly larger class of hypergroups. We further provide new necessary and sufficient conditions for strong Arens irregularity of $L^{1}(H)$ in terms of the spectral behaviour of $ν_{\infty}$, explore weighted (Beurling-type) hypergroup algebras, and obtain the first detailed comparison between the left and right topological centres for a wide class of non-classical examples. Several concrete applications to Jacobi, Naimark, and Bessel--Kingman hypergroups are presented. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_09955 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Arens Products and Asymptotic Structures on Chébli-Trimèche Hypergroups under Low Regularity Conditions Sababe, Saeed Hashemi Functional Analysis Primary 43A62, 43A20, Secondary 43A15, 46H20, 46B20 We investigate the Arens products on the second duals of convolution algebras associated with Chébli--Trimèche hypergroups, particularly focusing on the left and right topological centres of $L^{1}(H)^{\prime\prime}$ and $M(H)^{\prime\prime}$. Building on the recent framework established by Losert, we relax the classical smoothness assumptions on the underlying Sturm--Liouville function $A$ and develop new asymptotic analysis tools for measure-valued and low-regularity perturbations. This allows us to extend the existence and continuity of the asymptotic measures $ν_{x}$ and the limit measure $ν_{\infty}$ to a strictly larger class of hypergroups. We further provide new necessary and sufficient conditions for strong Arens irregularity of $L^{1}(H)$ in terms of the spectral behaviour of $ν_{\infty}$, explore weighted (Beurling-type) hypergroup algebras, and obtain the first detailed comparison between the left and right topological centres for a wide class of non-classical examples. Several concrete applications to Jacobi, Naimark, and Bessel--Kingman hypergroups are presented. |
| title | Arens Products and Asymptotic Structures on Chébli-Trimèche Hypergroups under Low Regularity Conditions |
| topic | Functional Analysis Primary 43A62, 43A20, Secondary 43A15, 46H20, 46B20 |
| url | https://arxiv.org/abs/2512.09955 |