  We construct an infinite dimensional non-unital Banach algebra $A$ and $a\in
A$ such that the sets $\{za^n:z\in\C,\ n\in\N\}$ and $\{({\bf
1}+a)^na:n\in\N\}$ are both dense in $A$, where $\bf 1$ is the unity in the
unitalization $A^{\#}=A\oplus \spann\{{\bf 1}\}$ of $A$. As a byproduct, we get
a hypercyclic operator $T$ on a Banach space such that $T\oplus T$ is
non-cyclic and $\sigma(T)=\{1\}$.
