The Fourier algebra of a rigid $C^{\ast }$-tensor category

Completely positive and completely bounded mutlipliers on rigid $C^{\ast }$-tensor categories were introduced by Popa and Vaes. Using these notions, we define and study the Fourier-Stieltjes algebra, the Fourier algebra and the algebra of completely bounded multipliers of a rigid $C^{\ast }$-tensor category. The rich structure that these algebras have in the setting of locally compact groups is still present in the setting of rigid $C^{\ast }$-tensor categories. We also prove that Leptin’s characterization of amenability still holds in this setting, and we collect some natural observations on property (T).