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?

  • $\begingroup$ This makes perfectly sense. $\endgroup$ – Jochen Jun 15 '14 at 10:44
  • 1
    $\begingroup$ In the non-unital case there is a better definition ... google for "non-degenerate character" or something like that. $\endgroup$ – Martin Brandenburg Jun 15 '14 at 11:05

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 $n\geq2$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.