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

TL;DR

This paper extends the concepts of Fourier and multiplier algebras from locally compact groups to rigid $C^{\

Contribution

It introduces and analyzes the Fourier-Stieltjes, Fourier, and multiplier algebras for rigid $C^{\

Findings

The structure of these algebras parallels that of locally compact groups.

Leptin's characterization of amenability applies in this setting.

Observations on property (T) are discussed.

Abstract

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).