Must a Banach algebra be unital in order to have a nonempty character space?

Question

The character or structure space $\Delta A$ of a Banach algebra $A$ over $\mathbb{C}$ is the set of all nonzero multiplicative homomorphisms of the form $f:A \to \mathbb{C}$.

I've only ever seen authors mention the space $\Delta A$ when talking about commutative, unital Banach algebras, and I'm curious if these properties are necessary for the space to be nontrivial.

Are there any examples of spaces with nonempty character spaces which are not unital?

Answer 1

No. Let $X$ be locally compact Hausdorff and not compact. For example, $X= \mathbb{R}$. Then $$C_0(X) \to \mathbb{C}: f \mapsto f(x)$$ is a character for all $x \in X$.

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If you want a non-commutative example, simply consider $$A \oplus \mathbb{C} \to \mathbb{C}: (a,\lambda) \mapsto \lambda$$ where $A$ is any non-commutative $C^*$-algebra.

Answer 2

Take $C_0((0,1])\equiv\{f\in C[0,1]: f(0)=0\}$. Under the natural operations and the supremum norm, this is a Banach algebra (a $C^*$-algebra actually). This is of course non unital, since if $fg=f$ for all $f\in C_0(0,1]$ we have that $g(t)=1$ for all $t\in[0,1]$, which is impossible if $g\in C_0(0,1]$ (since $g(0)=0$). But, if $t\in(0,1]$, then we have a naturally defined character $\phi_t:C_0(0,1]\to\mathbb{C}$ defined by evaluation at $t$, i.e. $\phi_t(f)=f(t)$.