 Scalar right action on a maximal-left-ideal quotient

In a complex unital Banach algebra, right multiplication by $a$ induces a bounded module endomorphism $T$ of $A/L$ when $La\subseteq L$. Every nonzero module endomorphism of this simple module is algebraically bijective, since its kernel and range are submodules. Its inverse is bounded by the bounded inverse theorem. Choosing $\lambda$ in the nonempty spectrum of an element of $T$ forces $T-\lambda I=0$. Applying this identity to $1+L$ gives $a-\lambda 1\in L$, with uniqueness because $1\notin L$.