Johnson pseudo-contractibility of certain Banach algebras and their nilpotent ideals

In this paper, we study the notion of Johnson pseudo-contractibility for certain Banach algebras. For a bicyclic semigroup $S$, we show that ${\mathrm{\ell }}^1(S)$ is not Johnson pseudo-contractible. Also for a Johnson pseudo-contractible Banach algebra $A$, we show that $A$ has no non-zero complemented closed nilpotent ideal.