Banach algebra 's homomorphic functional

Question

If $\mathscr{X}$ is a Banach algebra over $\mathbb{C}$, is it every homomorphism $\phi : \mathscr{X} \rightarrow \mathbb{C}$ continuous? $\mathscr{X}$ can have a identity and commutative if needed. Any helps will be appreciated.

Answer 1

First notice that, in case $\mathscr X$ has no unit, it is always possible to adjoin one and extend $\phi$ by setting $\phi(1)=1$. So we may assume that $\mathscr X$ is unital and that $\phi$ preserves units.

The kernel of $\phi$ is therefore a proper ideal which cannot contain any element $x$ with $\|x-1\|<1$ since all such elements are invertible.

This implies that $\text{Ker}(\phi)$ is not dense, and hence its closure is also a proper ideal.

However the kernel of a homomorphism is always a maximal ideal, hence $\text{Ker}(\phi)$ must coincide with its closure, i.e., it is closed.

It is well known that every linear functional with a closed kernel is continuous, hence $\phi$ is continuous.