# A note on left $\phi$-biflat Banach algebras

## TL;DR

This paper investigates the property of left $$-biflatness in Banach algebras, establishing its connection to amenability of groups and characterizing it for semigroup algebras, thus advancing understanding of algebraic structures in functional analysis.

## Contribution

It introduces and characterizes the concept of left $$-biflatness in Banach algebras, linking it to amenability and biflatness in specific algebraic contexts.

## Key findings

- Segal algebra $S(G)$ is left $$-biflat iff $G$ is amenable.
- Left $$-biflatness of $ell^{1}(S)$ characterized by biflatness of $S$.
- Provides new insights into the structure of Banach algebras related to group and semigroup properties.

## Abstract

In this paper, we study the notion of $\phi$-biflatness for some Banach algebras, where $\phi$ is a non-zero multiplicative linear functional. We show that the Segal algebra $S(G)$ is left $\phi$-biflat if and only if $G$ is amenable. Also, we characterize left $\phi$-biflatness of semigroup algebra $\ell^{1}(S)$ in the term of biflatness, where $S$ is a Clifford semigroup.