Johnson pseudo-Connes amenability of dual Banach algebras

TL;DR

This paper introduces Johnson pseudo-Connes amenability for dual Banach algebras, explores its properties, and characterizes it for group measure algebras and matrix algebras, linking it to classical amenability.

Contribution

It defines a new form of amenability for dual Banach algebras and investigates its properties and implications, including characterizations for specific algebra classes.

Findings

M(G) is Johnson pseudo-Connes amenable if and only if G is amenable

Finite index property for matrix algebras under this notion

Examples of dual Banach algebras exhibiting Johnson pseudo-Connes amenability

Abstract

We introduce the notion of Johnson pseudo-Connes amenability for dual Banach algebras. We study the relation between this new notion with the various notions of Connes amenability like Connes amenability, approximate Connes amenability and pseudo Connes amenability. We also investigate some hereditary properties of this new notion. We prove that for a locally compact group $G$, $M(G)$ is Johnson pseudo-Connes amenable if and only if $G$ is amenable. Also we show that for every non-empty set $I$, $\mathbb{M}_I(\mathbb{C})$ under this new notion is forced to have a finite index. Finally, we provide some examples of certain dual Banach algebras and we study their Johnson pseudo-Connes amenability.