# Pointwise amenability for dual Banach algebras

## TL;DR

This paper introduces and studies two new notions of pointwise amenability tailored for dual Banach algebras, focusing on their properties in specific sequence and semigroup algebras with respect to the $w^*$-topology.

## Contribution

It develops the concepts of pointwise Connes amenability and $w^*$-approximate Connes amenability for dual Banach algebras, analyzing their properties in weighted sequence and semigroup algebras.

## Key findings

- Characterization of pointwise amenability for $ell^1(
abla)$ and $ell^1(
abla,
abla)$
- Analysis of pointwise Connes amenability in weighted semigroup algebras
- Investigation of diagonals related to pointwise amenability in these algebras

## Abstract

We shall develop two notions of pointwise amenability, namely pointwise Connes amenability and pointwise $w^*$-approximate Connes amenability, for dual Banach algebras which take the $w^*$-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.