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

## TL;DR

This paper investigates the properties of left -biflatness and left -biprojectivity in Banach algebras, establishing conditions under which these properties hold for various algebraic constructions and their relation to the structure of groups.

## Contribution

It introduces new results linking left -biprojectivity of the second dual to left -biflatness of the algebra and characterizes these properties for matrix algebras and tensor products.

## Key findings

- Left -biprojectivity of $A^{**}$ implies left -biflatness of $A$.
- Characterization of -biprojectivity for $M(G)\u2297_{p}A(G)$ and $M(G)\u2297_{p}L^1(G)$ based on group properties.
- Conditions under which tensor products of Banach algebras are left -biprojective, depending on the finiteness or compactness of the group.

## Abstract

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.