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

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.