Let $A$ be a Banach algebra. Is there a Banach algebra $B$ and a non-trivial closed ideal $I$ of $B$ such that $\frac{B}{I}\cong A$?



Quotient of a Banach algebra
Sure. Let $K$ be any compact Hausdorff space that contains at least two points and for $B$ take the space of $A$ valued continuous functions on $K$. Take any $p$ in $K$. Let $I$ be the ideal of all functions in $B$ that vanish at $p$.


