On $\phi$-Biflatness-like properties of certain Banach algebras with applications

Abstract

In this paper left $\phi$-biflatness of abstract Segal algebras is investigated. For a locally compact group $G$, we show that any abstract Segal algebra with respect to $L^{1}(G)$ is left $\phi$-biflat if and only if the underlying group $G$ is amenable. We then prove that the Lipschitz algebras $\rm{Lip}_{\alpha}(X)$ and $\rm{lip}_{\alpha}(X)$ are left $C$-$\phi$-biflat if and only if $X$ is finite. Finally, we also study left $\phi$-biflatness of lower triangular matrix algebras.