When exactly is the character space of a Banach algebra empty?

Question

It is well-known that the character space, (i.e. the set of multiplicative characters) of a commutative, unital Banach algebra is non-empty.

But is there a complete characterization of when exactly the character space of Banach algebra is empty? Or even an incomplete characterization (ie certain classes of Banach algebras which we know have empty character space)?

Edit: I’d appreciate any responses, especially those with links for further reading!

Answer 1

Given a Banach algebra $B$ let $[B, B]$ (the commutator ideal) be the intersection of all kernels of morphisms of Banach algebras $B \to A$ where $A$ is commutative. This is a closed two-sided ideal of $B$, and the quotient $B/[B, B]$ is the abelianization of $B$; the universal commutative Banach algebra to which $B$ maps.

(Probably $[B, B]$ is the closure of the ideal generated by commutators $[b_1, b_2]$ but this construction of it manifestly has the appropriate universal property so it doesn't matter too much either way.)

Any character of $B$ must be a character of its abelianization, so the problem for arbitrary Banach algebras reduces immediately to the commutative case. A nonzero commutative Banach algebra always admits a character (by the Gelfand-Mazur theorem), so we get:

Proposition: $B$ admits a character iff $B/[B, B]$ is nonzero.

It's worth noting that $B/[B, B]$ can in fact be zero, for example if $B$ is simple and noncommutative (e.g. $B = M_n(\mathbb{C})$) as noted in the comments.