Fourier algebras of hypergroups and central algebras on compact (quantum) groups

TL;DR

This paper explores the properties of Fourier hypergroups and their algebras, establishing new characterizations of amenability, introducing weak amenability for hypergroups, and analyzing Fourier algebras of compact quantum groups.

Contribution

It introduces a notion of weak amenability for hypergroups, proves all discrete commutative hypergroups are weakly amenable with constant 1, and links Fourier algebras of hypergroups to quantum group centers.

Findings

Discrete commutative hypergroups are weakly amenable with constant 1

Provides a sufficient condition for hypergroup Fourier algebra amenability

Establishes isometric isomorphism between Fourier algebras of quantum groups and their centers

Abstract

This paper concerns the study of regular Fourier hypergroups through multipliers of their associated Fourier algebras. We establish hypergroup analogues of well-known characterizations of group amenability, introduce a notion of weak amenability for hypergroups, and show that every discrete commutative hypergroup is weakly amenable with constant 1. Using similar techniques, we provide a sufficient condition for amenability of hypergroup Fourier algebras, which, as an immediate application, answers one direction of a conjecture of Azimifard--Samei--Spronk [J. Funct. Anal. 256(5) 1544-1564, 2009] on the amenability of $ZL^1(G)$ for compact groups $G$. In the final section we consider Fourier algebras of hypergroups arising from compact quantum groups $\mathbb{G}$, and in particular, establish a completely isometric isomorphism with the center of the quantum group algebra for compact $\mathbb{G}$ of Kac type.