Inner amenability and approximation properties of locally compact quantum groups

We introduce an appropriate notion of inner amenability for locally compact quantum groups, study its basic properties, related notions, and examples arising from the bicrossed product construction. We relate these notions to homological properties of the dual quantum group, which allow us to generalize a well-known result of Lau--Paterson, resolve a recent conjecture of Ng--Viselter, and prove that, for inner amenable quantum groups $\mathbb{G}$, approximation properties of the dual operator algebras can be averaged to approximation properties $\mathbb{G}$. Similar homological techniques are used to prove that $\ell^1(\mathbb{G})$ is not relatively operator biflat for any non-Kac discrete quantum group $\mathbb{G}$; a discrete Kac algebra $\mathbb{G}$ with Kirchberg's factorization property is weakly amenable if and only if $L^1_{cb}(\widehat{\mathbb{G}})$ is operator amenable, and amenability of a locally compact quantum group $\mathbb{G}$ implies $C_u(\widehat{\mathbb{G}})=L^1(\widehat{\mathbb{G}})\widehat{\otimes}_{L^1(\widehat{\mathbb{G}})}C_0(\widehat{\mathbb{G}})$ completely isometrically. The latter result allows us to partially answer a conjecture of Voiculescu when $\mathbb{G}$ has the approximation property.Comment: v2: 32 pages. Several remarks and details added to improve the presentation. A few minor corrections. To appear in Indiana Univ. Math.

Fourier algebras of hypergroups and central algebras on compact (quantum) groups

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 [J. Funct. Anal. 256(5) 1544-1564, 2009] 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 $\mathbb{G}$, and in particular, establish a completely isometric isomorphism with the center of the quantum group algebra for compact $\mathbb{G}$ of Kac type.Comment: 23 page