  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).
