Pointwise amenability for dual Banach algebras

Abstract

We shall develop two notions of pointwise amenability, namely pointwise Connes amenability and pointwise $w^{\ast }$-approximate Connes amenability, for dual Banach algebras which take the $w^{\ast }$-topology into account. We shall study these concepts for the Banach sequence algebras ${\ell }^1(\omega )$ and the weighted semigroup algebras ${\ell }^1({\mathbb{N}}_{\wedge },\omega )$. For a weight $\omega$ on a discrete semigroup $S$, we shall investigate pointwise amenability/Connes amenability of ${\ell }^1(S,\omega )$ in terms of diagonals.

9 pages