Example of a non zero multiplicative linear functional on a non commutative Banach algebra

Question

Can anyone give an example of a non-commutative Banach Algebra with a non- zero multiplicative linear functional on it?

Answer 1

Take $A=\mathbb C\oplus M_2(\mathbb C)$, and put $\phi(a\oplus B)=a$.

$%\text{In a sense, the above example is the only kind of example possible. Because if $\phi$ is a character of $A$, then $J=\ker \phi$ is a closed two-sided ideal of $A$ such that $A/J$ is one-dimensional. Let $a\in A$ such that $\phi(a)=1$. Then as a vector space $A=J\oplus\mathbb C a$, since every $b\in A$ can be written as $b=(b-\phi(b)a)+\phi(b)a $. }$

$%\text{We may assume that $a=a^*$, since $\phi(a^*)=\overline{\phi(a)}=1$, and so $\phi((a+a^*)/2)=1$. And then we may assume that $a\geq0$, because if $a^+\in J$ then $\phi(a)=\phi(a^-)\leq0$. }$