Arens Products and Asymptotic Structures on Chébli-Trimèche Hypergroups under Low Regularity Conditions

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.

💡 Research Summary

This paper extends the theory of Arens products on the biduals of convolution algebras associated with Chébli‑Trimèche hypergroups to a much broader class of hypergroups by weakening the regularity assumptions on the underlying Sturm–Liouville coefficient A. Traditionally, analyses of Arens products and topological centres for hypergroup algebras required A to be at least C² with well‑behaved derivatives. The author replaces this by a minimal condition (SL′): A∈C¹ on (0,∞) and its derivative A′ is of bounded variation (or, more generally, a bounded Radon measure) on every compact interval. This permits jump discontinuities, piecewise smooth behaviour, and measure‑valued perturbations.

The first technical achievement is the construction of eigenfunctions of the Sturm–Liouville operator L f=−(1/A)(A f′)′ under (SL′). By introducing the phase function Φ(x)=∫_{x₀}^{x}A(t)^{-½}dt and writing solutions as u(x)=e^{iλΦ(x)}m(x,λ), the differential equation is transformed into a Volterra integral equation for m. The kernel K(x,t;λ) and the measure µ (derived from A′) satisfy uniform integrability bounds that depend linearly on the total variation of A′. Lemma 3.2 and Lemma 3.3 prove existence, uniqueness, and uniform closeness of m to 1, using a Neumann series argument that works as long as the variation of A′ on the tail is sufficiently small—a condition guaranteed by the growth of A.

With these eigenfunctions, the author defines asymptotic convolution measures ν_{x,y}=δ_{−y}∗(δ_x∗δ_y) for y>x and shows, in Theorem 3.4, that ν_{x,y} converges in L¹(ℝ) (or weak‑∗) to a limit ν_x as y→∞. Moreover, the map x↦ν_x is continuous, and ν_x itself converges to a limit ν_∞ as x→∞. These measures capture the “far‑field” behaviour of the hypergroup convolution and are the key objects governing the Arens product structure.

The central new result is a necessary and sufficient condition for strong Arens irregularity of L¹(H). The author proves that L¹(H) is strongly Arens irregular (i.e., the left topological centre Z_t(L¹(H)″) coincides with the canonical copy of L¹(H)) if and only if the Fourier transform of ν_∞ does not vanish identically. In symbols, \