Essential character amenability of Banach algebras

For a Banach algebra ${\cal A}$ and a character $\phi$ on ${\cal A}$, we introduce and study the notion of essential $\phi$-amenability of ${\cal A}$. We give some examples to show that the class of essentially $\phi$-amenable Banach algebras is larger than that of $\phi$-amenable Banach algebras introduced by Kaniuth, Lau and Pym. Finally, we characterize essential $\phi$-amenability of various Banach algebras related to locally compact groups