Must a Banach algebra be unital and commutative in order to have characters?

Question

‎Let $A$ be a Banach algebra.

1. Is ‎there a‎n ‎abelian ‎Banach ‎algebra ‎$A$ without ‎identity ‎so ‎that‎ ‎$\Omega (‎A)=‎\varnothing$‎‎? ‎

2. ‎Is ‎there a‎ Banach ‎algebra ‎with ‎identity $A$ ‎ ‎so ‎that‎ ‎$\Omega (A)=‎\varnothing$?‎

I would like to know whether ‎the ‎abelian ‎and ‎identity ‎are ‎necessary ‎to ‎have $\Omega (A)\neq‎‎ \varnothing $‎, where

‎$$\Omega(A) = \{ \varphi \colon A \longrightarrow \mathbb{C}: \varphi \text{ is a non-zero homomorphism}\}$$

Answer 1

As for your first question, yes, there are such algebras. Please see here.

In the non-abelian case characters are somehow rare. For example, for every Banach space $X$ isomorphic to $X\oplus X$, the unital Banach algebra $B(X)$ comprising all bounded linear operators on $X$ does not have any characters because $B(X)$ is Banach-algebra isomorphic to every matrix algebra $M_n(B(X))$ over itself.