A note on essentially left $\phi$-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 $\phi$-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 $\phi$-contractible. We discuss essential left $\phi$-contractibility of some Fourier algebras.