Strong pseudo-Connes amenability of dual Banach algebras

TL;DR

This paper introduces the concept of strong pseudo-Connes amenability for dual Banach algebras, explores its relation to existing notions, and examines its implications for matrix algebras, semigroup algebras, and various Banach algebra constructions.

Contribution

It defines a new form of amenability for dual Banach algebras and investigates its properties and applications across different algebraic structures.

Findings

$M_I(C)$ is strong pseudo-Connes amenable iff $I$ is finite

${ m ext{ extltilde}}^{1}(S)^{**}$ has an ultra central approximate identity iff $S$ is a group

The property of ultra central approximate identity is studied for various Banach algebra constructions

Abstract

In this paper, we introduce the new notion of strong pseudo-Connes amenability for dual Banach algebras. We study the relation between this new notion to the various notions of Connes amenability. Also we show that for every non-empty set $I$, $M_I(\mathbb{C})$ is strong pseudo-Connes amenable if and only if $I$ is finite. We provide some examples of certain dual Banach algebras and we study its strong pseudo-Connes amenability. In the last section, we investigate the property ultra central approximate identity for a Banach algebra $\mathcal{A}$ and its second dual $\mathcal{A}^{**}$. Also we show that for a left cancellative regular semigroup $S$, ${\ell^{1}(S)}^{**}$ has an ultra central approximat identity if and only if $S$ is a group. Finally we study this property for $\varphi$-Lau product Banach algebras and the module extension Banach algebras.