Question on representation of a Banach algebra

Question

Let $A$ be a Banach algebra and $\pi$ a continuous irreducible representation of $A$ on a Banach space $X$. Suppose $\pi(a)\xi\neq0$ for some $a\in A$ and some $\xi\in X$. Let $\eta\in X$. The question is to show that there exists $b\in A$ such that $\pi(b)\pi(A)\xi=\eta$. We can of course find some linear operator $T$ on $X$ such that $T(\pi(a)\xi)=\eta$, but how do we ensure that $T$ is of the form $\pi(b)$ for some $b\in A$?

Answer 1

Consider the image of the left ideal $Aa$ under $\pi$. By irreducibility, $\pi(A) \pi(a) \xi = X$. So you're done.