Biflat-like Banach algebras

Abstract

Given a Banach algebra $\mathcal{A}$ and a continuous homomorphism $\sigma$ on it, the notion of $\sigma$-biflatness for $\mathcal{A}$ is introduced. This is a generalization of biflatness and it is shown that they are distinct. The relations between $\sigma$-biflatness and some other close concepts such as $\sigma$-biprojectivity and $\sigma$-amenability are studied. The $\sigma$-biflatness of tensor product of Banach algebras are also discussed.