Varvara Shepelska

Weak Amenability Of Certain Beurling Algebras

Date Friday, January 24, 2014

For a locally compact Abelian group $G$ weak amenability of the weighted algebra $L^1(G,omega)$ was recently characterized by Yong Zhang. In this talk I will discuss weak amenability of $L^1(G,omega)$ for some non-commutative groups $G$. My main focus will be on the group $mathbb{F}_2$ - the free group on two generators, which is one of the simplest non-Abelian groups. I will consider several natural classes of weights on $mathbb{F}_2$ and show that if a weight $omega$ belongs to one of these classes, then the algebra $ell^1(mathbb{F}_2,omega)$ can only be weakly amenable when $omega$ is diagonally bounded, that is $sup_{xinmathbb{F}_2}omega(x)omega(x^{-1})<infty$. In particular, this will give a rise to an example showing that the necessary and sufficient condition of Y. Zhang for weak amenability of an Abelian algebra $L^1(G,omega)$ is no longer sufficient for $ell^1(mathbb{F}_2,omega)$, which means that the situation for non-commutative groups is completely different. I will also present some necessary conditions on the weight $omega$ for the algebra $L^1(G,omega)$ to be weakly amenable in case when the group $G$ is solvable, or $G$ is an $[IN]$ group.