On left φ-biprojectivity and left φ-biflatness of certain Banach algebras

In this paper, we study left $\phi$-biflatness and left $\phi$-biprojectivity of some Banach algebras, where $\phi$ is a non-zero multiplicative linear function. We show that if the Banach algebra $A^{**}$ is left $\phi$-biprojective, then $A$ is left $\phi$-biflat. Using this tool we study left $\phi$-biflatness of some matrix algebras. We also study left $\phi$-biflatness and left $\phi$-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 $\phi\otimes \psi$-biprojective if and only if $G$ is finite. We show that $M(G)\otimes_{p} L^1(G)$ is left $\phi\otimes \psi$-biprojective if and only if $G$ is compact.