Connes-biprojective dual Banach algebra

TL;DR

This paper introduces the concept of Connes-biprojectivity for dual Banach algebras, explores its relation to Connes-amenability, and establishes conditions under which dual and Arens regular Banach algebras exhibit this property.

Contribution

It defines Connes-biprojectivity for dual Banach algebras and links it to Connes-amenability, providing new insights into the structure of these algebras.

Findings

Connes-amenability is equivalent to Connes-biprojectivity plus having a bounded approximate identity.

For Arens regular biprojective Banach algebras, their second duals are Connes-biprojective.

Introduces a new notion of biprojectivity tailored for dual Banach algebras.

Abstract

In this paper, we introduce a new notion of biprojectivity, called Connes-biprojective, for dual Banach algebras. We study the relation between this new notion to Connes-amenability and we show that, for a given dual Banach algebra $ \mathcal{A} $, it is Connes-amenable if and only if $ \mathcal{A} $ is Connes-biprojective and has a bounded approximate identity. Also, for an Arens regular Banach algebra $ \mathcal{A} $, we show that if $ \mathcal{A} $ is biprojective, then the dual Banach algebra $ \mathcal{A} ^{**} $ is Connes-biprojective.