Characters only on commutative unital algebras?

Question

I saw the following definition of a character:

Let $A$ be a commutative unital Banach algebra. Then a non-zero homomorphism $\chi : A \to \mathbb C$ is called a character.

For this definition to make sense does $A$ have to be commutative? It seems to me that it shouldn't make much of a difference for the homomorphism whether $A$ is commutative or not.

Similarly, it's not quite clear to me if $A$ really has to be unital.

Is it possible to define characters for a not necessarily commutative, not necessarily unital Banach algebras $A$ to be homomorphisms $A \to \mathbb C$ or would it not make sense?

Answer 1

You can certainly consider nonzero homomorphisms $A\to\mathbb C$ for any Banach algebra $A$. The thing is that they often don't exist: for example $M_n(\mathbb C) $ has no characters for any $n\geq2$.