= Scalar right action on a maximal-left-ideal quotient
{title2=$La\subseteq L\Longrightarrow a-\lambda1\in L$}
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 $a-\lambda1\in L$, with uniqueness because $1\notin L$.
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