Character on Unital Banach Algebra is Uniquely Identified by Kernel

Theorem
Let $\struct {A, \norm {\, \cdot \,} }$ be a unital Banach algebra over $\C$.

Let $\phi, \psi : A \to \C$ be characters on $A$ such that:
 * $\ker \phi = \ker \psi$

Then $\phi = \psi$.

Proof
From Character on Unital Banach Algebra is Unital Algebra Homomorphism, we have $\map \phi { {\mathbf 1}_A} = 1$ and $\map \psi { {\mathbf 1}_A} = 1$.

Let $x \in A$.

We have:

So we have $x - \map \phi x {\mathbf 1}_A \in \ker \phi$.

Since $\ker \phi = \ker \psi$, we have $x - \map \phi x {\mathbf 1}_A \in \ker \psi$.

So, we have:
 * $\map \psi {x - \map \phi x {\mathbf 1}_A} = 0$

We therefore obtain that:

So we have $\map \phi x = \map \psi x$.

Since $x \in A$ was arbitrary, we obtain $\phi = \psi$.