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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?

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  • $\begingroup$ This makes perfectly sense. $\endgroup$
    – Jochen
    Jun 15, 2014 at 10:44
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    $\begingroup$ In the non-unital case there is a better definition ... google for "non-degenerate character" or something like that. $\endgroup$
    – Martin Brandenburg
    Jun 15, 2014 at 11:05

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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 $n\geq2$.

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