On $\phi$-biflat and $\phi$-biprojective Banach algebras

Abstract

In this paper, we introduce the new notions of $\phi$-biflatness, $\phi$-biprojectivity, $\phi$-Johnson amenability and $\phi$-Johnson contractibility for Banach algebras, where $\phi$ is a non-zero homomorphism from a Banach algebra $A$ into $\mathbb{C}$. We show that a Banach algebra $A$ is $\phi$-Johnson amenable if and only if it is $\phi$-inner amenable and $\phi$-biflat. Also we show that $\phi$-Johnson amenability is equivalent with the existence of left and right $\phi$-means for $A$. We give some examples to show differences between these new notions and the classical ones. Finally, we show that ${L^{1}(G)}$ is $\phi$-biflat if and only if $G$ is an amenable group and $A(G)$ is $\phi$-biprojective if and only if $G$ is a discrete group.