Character on Non-Unital Banach Algebra induces Character on Unitization

Theorem
Let $\struct {A, \norm {\, \cdot \,} }$ be a Banach algebra over $\C$ that is not unital.

Let $\struct {A_+, \norm {\, \cdot \,}_{A_+} }$ be the normed unitization of $\struct {A, \norm {\, \cdot \,} }$.

Let $\phi$ be a character on $A$.

Define:
 * $\map {\phi_+} {\tuple {x, \lambda} } = \map \phi x + \lambda$

for each $\tuple {x, \lambda} \in A_+$.

Then $\phi_+$ is a character on $A_+$.

Proof
Let $\tuple {x, \lambda}, \tuple {y, \mu} \in A_+$ and $t \in \C$.

We have:

and so $\phi_+$ is linear.

To show that $\phi_+$ is a character, it remains to show that:
 * $\map {\phi_+} {\tuple {x, \lambda} \tuple {y, \mu} } = \map {\phi_+} {\tuple {x, \lambda} } \map {\phi_+} {\tuple {y, \lambda} }$

We have:

So $\phi_+$ is a character.