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

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).

arXiv comment13 pages; this article contains the material from Sections 3, 4 and 7 from the second version of arXiv:1605.08658, to appear in Publications of the RIMS

References in corpus (8)

• CCAP for universal discrete quantum groups
• Simple Lie groups without the Approximation Property
• Simple Lie groups without the Approximation Property II
• Representation theory for subfactors, $\lambda$-lattices and C*-tensor categories
• Around Property (T) for quantum groups
• C*-tensor categories and subfactors for totally disconnected groups
• Howe-Moore type theorems for quantum groups and rigid C*-tensor categories
• Approximation properties for $p$-adic symplectic groups and lattices

Cited by in corpus (4)

• Property (T) discrete quantum groups and subfactors with triangle presentations
• Pointwise convergence of noncommutative Fourier series
• The approximation property for locally compact quantum groups
• $L^2$-Betti numbers of $C^{\ast }$-tensor categories associated with totally disconnected groups