In this article we study the Fourier space of a general hypergroup and its multipliers. The main result of this paper characterizes commutative hypergroups whose Fourier space forms a Banach algebra under pointwise product with an equivalent norm. Among those hypergroups whose Fourier space forms a Banach algebra, we identify a subclass for which the Gelfand spectrum of the Fourier algebra is equal to the underlying hypergroup. This subclass includes for instance, Jacobi hypergroups, Bessel-Kingman hypergroups.2000 Mathematics subject classification: primary 43A62; secondary 43A22,43A10, 46J10.

We in this article, introduce a class of hypergroups called ultraspherical hypergroups and show that the Fourier space of an ultraspherical hypergroup forms a Banach algebra under pointwise product. These hypergroups need not be commutative and include for example double coset hypergroups. We also show that the structure space of this algebra equals the underlying hypergroup. IntroductionFourier algebras of locally compact groups play an important role in the field of harmonic analysis on locally compact groups and are being extensively studied as is suggested for instance, by the survey article of Eymard [8]. But their counterparts defined on hypergroups do not enjoy the same prominence, since they often fail to be algebras under pointwise product. One important reason is that the product of two continuous positive definite functions on a hypergroup need not be positive definite in general. Yet, the existence of several classes of examples of hypergroups like Jacobi hypergroups for example, for which the product of positive definite functions belonging to the support of Plancherel-Levitan measure is again positive definite, motivates us to study those hypergroups for which the Fourier space forms a Banach algebra.In the article Muruganandam [15], we develop a general theory of Fourier spaces of hypergroups and give necessary and sufficient conditions for a commutative hypergroup to have an equivalent norm with respect to which the Fourier space forms a Banach algebra under pointwise product.In our attempt to obtain examples of noncommutative hypergroups whose Fourier space is a Banach algebra, we in this article, which is subsequent to the one cited above, introduce a new class of hypergroups called spherical hypergroups and a subclass namely ultraspherical hypergroups which includes for example double coset hypergroups. We show that Fourier spaces of ultraspherical hypergroups form Banach algebras under pointwise product. We also show that the structure space consisting of all nonzero complex homomorphisms of this algebra is homeomorphic to the underlying hypergroup.

Analyzing phase transitions using the inherent geometrical attributes of a system has garnered enormous interest over the past few decades. The usual candidate often used for investigation is graphene- the most celebrated material among the family of tri co-ordinated graphed lattices. We show in this report that other inhabitants of the family demonstrate equally admirable structural and functional properties that at its core are controlled by their topology. Two interesting members of the family are Cylooctatrene(COT) and COT-based polymer: poly-bi-[8]-annulenylene both in one and two dimensions that have been investigated by polymer chemists over a period of 50 years for its possible application in batteries exploiting its conducting properties. A single COT unit is demonstrated herein to exhibit topological solitons at sites of a broken bond similar to an open one-dimensional Su-Schrieffer-Heeger (SSH) chain. We observe that Poly-bi-[8]-annulenylene in 1D mimics two coupled SSH chains in the weak coupling limit thereby showing the presence of topological edge modes. In the strong coupling limit, we investigate the different parameter values of our system for which we observe zero energy modes. Further, the application of an external magnetic field and its effects on the band-flattening of the energy bands has also been studied. In 2D, poly-bi-[8]-annulenylene forms a square-octagon lattice which upon breaking time-reversal symmetry goes into a topological phase forming noise-resilient edge modes. We hope our analysis would pave the way for synthesizing such topological materials and exploiting their properties for promising applications in optoelectronics, photovoltaics, and renewable energy sources.