A note on essentially left $\varphi$-contractible Banach algebras

Abstract

In this note, we show that \cite[Corollary 3.2]{sad} is not always true. In fact, we characterize essential left $\varphi$-contractibility of the the group algebras in the term of compactness of its related locally compact group. Also we show that for any compact commutative group $G$, $L^2(G)$ is always essentially left $\varphi$-contractible. We discuss essential left $\varphi$-contractibility of some Fourier algebras.