Suppose I have a unital Banach algebra $A$ and a maximal left ideal $L$ and an element $a$ such that $La\subset{}L$. I want to show that there exists (uniqueness is easy) some complex $\lambda$ such that $a-\lambda{}1$ is in $L$.
So far I have a feeling its something to do with the spectrum. I tried proving that if $a$ is not invertible then $a$ must be in $L$ which would be enough. I'm not sure if this is even true though. Thanks in advance!