$WAP$-biprojectivity of the enveloping dual Banach algebras

TL;DR

This paper introduces $WAP$-biprojectivity for enveloping dual Banach algebras, explores its relation to Connes properties, and provides examples and non-examples involving compact groups.

Contribution

It defines a new notion of biprojectivity, establishes its relation to Connes biprojectivity and amenability, and analyzes specific cases for dual Banach algebras.

Findings

Connes amenability of $F(\mathcal{A})$ implies Connes amenability of $\mathcal{A}$

$F(L^2(G))$ is not $WAP$-biprojective for infinite compact groups

Examples illustrating $WAP$-biprojectivity and Connes amenability of dual Banach algebras

Abstract

In this paper, we introduce a new notion of biprojectivity, called $WAP$-biprojectivity for $F(\mathcal{A})$, the enveloping dual Banach algebra associated to a Banach algebra $\mathcal{A}$. We find some relations between Connes biprojectivity, Connes amenability and this new notion. We show that, for a given dual Banach algebra $\mathcal{A}$, if $F(\mathcal{A})$ is Connes amenable, then $\mathcal{A}$ is Connes amenable. For an infinite commutative compact group $G$, we show that the convolution Banach algebra $F(L^2(G))$ is not $WAP$-biprojective. Finally, we provide some examples of the enveloping dual Banach algebras and we study their $WAP$-biprojectivity and Connes amenability.