Amenability and co-amenability in non-abelian group duality

Amenability and weak amenability of the Fourier algebra

Let $G$ be a locally compact group. We show that its Fourier algebra $A(G)$ is amenable if and only if $G$ has an abelian subgroup of finite index, and that its Fourier-Stieltjes algebra $B(G)$ is amenable if and only if $G$ has a compact, abelian subgroup of finite index. We then show that $A(G)$ is weakly amenable if the component of the identity of $G$ is abelian, and we prove some partial results towards the converse.

Weak amenability of Fourier algebras and local synthesis of the anti-diagonal

We show that for a connected Lie group $G$, its Fourier algebra $A(G)$ is weakly amenable only if $G$ is abelian. Our main new idea is to show that weak amenability of $A(G)$ implies that the anti-diagonal, $\checkΔ_G=\{(g,g^{-1}):g\in G\}$, is a set of local synthesis for $A(G\times G)$. We then show that this cannot happen if $G$ is non-abelian. We conclude for a locally compact group $G$, that $A(G)$ can be weakly amenable only if it contains no closed connected non-abelian Lie subgroups. In particular, for a Lie group $G$, $A(G)$ is weakly amenable if and only if its connected component of…

Amenability properties of Fourier algebras and Fourier-Stieltjes algebras: a survey

Let G be a locally compact group, and let A(G) and B(G) denote its Fourier and Fourier-Stieltjes algebras. These algebras are dual objects of the group and measure algebras, L^1(G) and M(G), in a sense which generalizes the Pontryagin duality theorem on abelian groups. We wish to consider the amenability properties of A(G) and B(G) and compare them to such properties for L^1(G) and M(G). For us, ``amenability properties'' refers to amenability, weak amenability, and biflatness, as well as some properties which are more suited to special settings, such as the hyper-Tauberian property for semisi…

Amenability and covariant injectivity of locally compact quantum groups II

Building on our previous work, we study the non-relative homology of quantum group convolution algebras. Our main result establishes the equivalence of amenability of a locally compact quantum group $\mathbb{G}$ and 1-injectivity of $L^{\infty}(\widehat{\mathbb{G}})$ as an operator $L^1(\widehat{\mathbb{G}})$-module. In particular, a locally compact group $G$ is amenable if and only if its group von Neumann algebra $VN(G)$ is 1-injective as an operator module over the Fourier algebra $A(G)$. As an application, we provide a decomposability result for completely bounded $L^1(\widehat{\mathbb{G}}…

关于 Fourier 代数的弱顺从性

On weak amenability of Fourier algebras

本文将连通 Lie 群 Fourier 代数弱顺从当且仅当群交换的结论，推广至一般连通局部紧群，扩展了 Choi 和 Ghandehari 的特殊情形方法。

紧量子群与离散量子群的余理想之融合模与顺从性

Fusion modules and amenability of coideals of compact and discrete quantum groups

本文定义了融合代数上的顺从融合模，利用相对可积性构造拟正则表示，刻画了其余顺从性质，推广了Tomatsu定理，并证明了非标准Podleś球面相关融合模的顺从性。