An abelian Banach algebra with no nonzero morphisms $A \to \mathbb{C}$

Question

We know that if $A$ is a abelian Banach algebra with identity, then the set ‎$\Omega ‎(A)$‎‎‎‎‎ of nonzero homomorphisms $\phi : A \to \mathbb{C}$ isn't empty.

Can you give an example of an abelian Banach algebra without identity $A$ such that $‎\Omega ‎(A) = \varnothing$?

Answer 1

$$\mathcal A:=\mathbb C$$ with $$a+_{\mathcal A}b:= a+_{\mathbb C}b\qquad a\cdot_{\mathcal A}b:=0\qquad\|a\|_{\mathcal A}:=|a|_{\mathbb C}$$

I'll leave it to you to verify that this is an abelian Banach algebra. For any character $\varphi$ you have $\varphi(a)^2=\varphi(a^2)=0$ for all $a$, so $\varphi(a)=0$.