On left $\varphi$-biprojectivity and left $\varphi$-biflatness of certain Banach algebras

Abstract

In this paper, we study left $\varphi$-biflatness and left $\varphi$-biprojectivity of some Banach algebras, where $\varphi$ is a non-zero multiplicative linear function. We show that if the Banach algebra $A^{\ast \ast }$ is left $\varphi$-biprojective, then $A$ is left $\varphi$-biflat. Using this tool we study left $\varphi$-biflatness of some matrix algebras. We also study left $\varphi$-biflatness and left $\varphi$-biprojectivity of the projective tensor product of some Banach algebras. We prove that for a locally compact group $G$, $M(G){\otimes }_{p}A(G)$ is left $\varphi \otimes \psi$-biprojective if and only if $G$ is finite. We show that $M(G){\otimes }_p{L}^1(G)$ is left $\varphi \otimes \psi$-biprojective if and only if $G$ is compact.