Norm of a character in a non-unital Banach algebra

Question

Let $\cal A$ be an abelian non-unital Banach algebra and $h:{\cal A}\to {\Bbb C}$ be a homomorphism. If ${\cal A}$ has an approximate identity $\{e_i\}$ such that $||e_i||\leq 1$ for all i, then $||h||=1.$

For this I can show that $||h||\leq1$, but I can not show $||h||=1.$ Please help me. $\infty$ Thanks.

Answer 1

I think you have already done the hard part. Assume $h \neq 0$ and fix some $a \in \mathcal{A}$ with $h(a) \neq 0$. For each $i$, we have $h(e_i) h(a) = h(e_i a)$, so $$ h(e_i) = \frac{h(e_ia)}{h(a)} \to 1.$$ Thus there is a net of elements of norm $\leq 1$ whose images tend to $1$. It follows that $\|h\| \geq 1$.