On approximate Connes-biprojectivity of dual Banach algebras

TL;DR

This paper introduces the concept of approximate Connes-biprojectivity for dual Banach algebras, explores its relations with other properties, and characterizes when certain algebras are approximately Connes-biprojective.

Contribution

It defines approximate Connes-biprojectivity, establishes criteria for dual triangular Banach algebras, and characterizes this property for group measure algebras and $L^2$ spaces.

Findings

$M(G)$ is approximately Connes-biprojective iff $G$ is amenable

Certain dual triangular Banach algebras are not approximately Connes-biprojective

For an infinite compact group $G$, $L^2(G)$ is approximately Connes-biprojective but not Connes-biprojective.

Abstract

In this paper, we introduce a notion of approximate Connes-biprojectivity for dual Banach algebras. We study the relation between approximate Connes-biprojectivity, Johnson pseudo-Connes amenability and $\varphi$-Connes amenability. We propose a criterion to show that some certain dual triangular Banach algebras are not approximately Connes-biprojective. Next we show that for a locally compact group $G$, the Banach algebra $M(G)$ is approximately Connes-biprojective if and only if $G$ is amenable. Finally for an infinite commutative compact group $G$ we show that the Banach algebra $L^2(G)$ with convolution product is approximately Connes-biprojective, but it is not Connes-biprojective.