Uniqueness of non-zero homomorphism in abelian Banach algebra

Question

Consider $\mathbb{C}^3$ equipped with product $\bullet$ given as $$(x_1,x_2,x_3)\bullet(y_1,y_2,y_3)~=~ (x_1y_1,x_1y_2+x_2y_1,x_1y_3+x_2y_2+x_3y_1)$$ and norm $\|(x_1,x_2,x_3)\|=|x_1|+|x_2|+|x_3|$. $A:=(\mathbb{C}^3,\bullet,\|\!\cdot\!\|)$ is then a unital abelian Banach algebra with unit $(1,0,0)$.

How would one verify that there cannot be other non-zero homomorphism on $A$ than $$A\ni(x_1,x_2,x_3)\mapsto x_1\in\mathbb{C}?$$

Answer 1

Let $f:A\to \Bbb{C}$ be an algebra homomorphism with $f(e_1) =1$

Then $f(e_2)•f(e_3)=f(e_2•e_3) =f(0) =0$

And $f(e_2^2) =f(e_3) $

Hence $f(e_2) =0=f(e_3) $