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 [7] 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 G, and in particular, establish a completely isometric isomorphism with the center of the quantum group algebra for compact G of Kac type.

Remarks on weak amenability of hypergroups

“…As the predual of V N (H), A(H) has a canonical operator space structure. A hypergroup H is called a completely Fourier hypergroup if A(H), furnished with its canonical operator space structure, is a completely contractive Banach algebra (look at [4,Section 3]). Based on this operator space structure, the space of completely bounded multipliers of A(H), denoted by M cb A(H), for a completely Fourier hypergroup H is defined.…”

“…For more on hypergroup Fourier and reduced Fourier-Stieltjes spaces, in particular on commutative and ultraspherical hypergroups, we refer the reader to [1,4,15,16].…”

“…In [4], it was shown that u → u/χ 0 forms an isometry from A(H) onto A(H 0 ) as Banach spaces. In fact, it was observed that A(H 0 ) is isometric to the closure of A(H) in its multiplier norm.…”

“…The spaces of bounded multipliers and then later completely bounded multipliers of hypergroup Fourier algebras have been studied in [4,15,16]. The notions of weak amenability and subsequently the Cowling-Haagerup constant of a hypergroup were introduced in [4] where Crann and the author showed that for every discrete commutative hypergroup, this constant is 1.…”

“…Pointing out the role of this double coset structure in studying the Cowling-Haagerup constants of Sp(n, 1) and F 4(−20) ([14]), Eymard ([11]) asks for the analogues of the completely bounded multipliers for the Fourier algebra and the Cowling-Haagerup constant for H α,β . He was in particular interested to know, if for H α,β , such a constant would change by parameters α, β.The spaces of bounded multipliers and then later completely bounded multipliers of hypergroup Fourier algebras have been studied in [4,15,16]. The notions of weak amenability and subsequently the Cowling-Haagerup constant of a hypergroup were introduced in [4] where Crann and the author showed that for every discrete commutative hypergroup, this constant is 1.…”