Example of Banach algebra with non-zero anhillator ideal

Question

Is it possible for Banach algebras to have non-zero anhillator? In other words, does there exist a Banach algebra $A$ with an element $a$ such that $ab=0$ for all $b\in A$ ? Please mention few examples if possible

Answer 1

Unless I'm missing something; what about the algebra of $2\times 2$ matrices (viewed as the algebra of bounded linear operators from $\mathbb{F}^2$ to itself) of the form $$ \left( \begin{array}{cc} a & b \\ 0 & 0 \end{array} \right), \qquad a,b\in \mathbb{F}. $$ Then every element with $a=0$ annihilates every other element.