If $\ell^1(S)$ is unital, must the semigroup $S$ have an identity?

Question

Assume $S$ is non-empty. If $\ell^1(S)$ is unital, then there exists $e\in\ell^1(S)$ with $f\star e=e\star f=f$ for all $f\in\ell^1(S)$ and $\|e\|_1=1$. Let $a\in S$. Then $\delta_a\in\ell^1(S)$ so that $$(\delta_a\star e)(a)=\sum_{as=a}e(s)=\delta_a(a)=1.$$ So there must exist $s\in S$ with $as=a$. Similarly, from the other side there must exist $t\in S$ with $ta=a$. I'm not really sure how to proceed from here, or if the result is even true.