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?