Character on Banach Algebra is Continuous/Corollary

Corollary

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

Let $\phi : A \to \C$ be a character on $A$.

Then:

$\norm \phi_{A^\ast} = 1$

Proof

From Character on Banach Algebra is Continuous, we have that $\phi$ is continuous with $\norm \phi_{A^\ast} \le 1$.

So, we have:

$\ds \sup_{x \in A, \, \norm x = 1} \cmod {\map \phi x} \le 1$

We have that $\norm { {\mathbf 1}_A} = 1$ and $\map \phi { {\mathbf 1}_A} = 1$, so we have:

$\ds \sup_{x \in A, \, \norm x = 1} \cmod {\map \phi x} = 1$

So $\norm \phi_{A^\ast} = 1$.

$\blacksquare$