Local banach algebra without zero divisors

Question

I need to construct example of banach algebra with unique nontrivial maximal ideal and without zero divisors. I think it is must be a subalgebra of $\mathbb{C}[[z]]$, but I could not build anything.

Answer 1

Let $A = \mathbb{C}[[t]]$ be the algebra of formal series. Let $B \subset A$ be some subalgebra, satisfying $$\| a \| = \sum{a_{n} |w_{n}|} < \infty$$ Here $w_{n}$ is a sequence of positive numbers that we actually need to find.

First, note that $B$ is indeed local, since the set of all non-invertivle elements of $A$ is an ideal. (update: why???)

How to find $w_{n}$?

Since the $A$ is a Banach algebra, then $$||ab|| \leq ||a||||b||$$ so by comparing term-by-term coefficients one can extract the desired values.

update: if one takes $w_{n} = (1, 1, \frac{1}{2!}, \ldots, \frac{1}{n!}, \ldots, )$, then the property above holds and, moreover, it looks as if the subalgebra is indeed local but i can't see a fast way to recover it.