Maximal ideals of the ring $\mathbb C \{T\}$ Consider the Banach $\mathbb C$-algebra
$$
\mathbb C \{T\} = \left\lbrace \sum_{i \geq 0} a_i T^i : \sum_{i \geq 0} |a_i| < \infty \right\rbrace
$$
With the norm given by $\| \sum a_i T^i\| = \sum |a_i|$. By the Gelfand-Mazur theorem, the maximal ideals of this ring is in correspondence with the set of multiplicative functionals $\chi : \mathbb C \{T\} \to \mathbb C$. It is not hard to see that such a functional must be given by
$$
\chi \left(\sum_{i \geq 0} a_i T^i \right)= \sum_{i \geq 0} a_i z^i
$$
for some $z$ with $|z| \leq 1$, so $\operatorname{Max} \mathbb C \{T\} = \overline{D(0,1)}$.
My questions are:

*

*Is there a way to see the last claim using a combination of commutative algebra and complex analysis, without directly invoking the Gelfand correspondence?

*Is there a source where algebraic properties of this sort of rings are studied? Like, a description of the prime ideals of $\mathbb C \{T_1, \ldots , T_n\}$, its Krull dimension etc. similar to the study of the local analytic rings $\mathcal O_{\mathbb C^n}$. I came across this question while learning Berkovich spaces, where Banach rings of the form $k \{T_1, \ldots, T_n\}$ are studied, but the theory there seems to depend heavily on the fact that $k$ is non-Archimedean.

EDITS:

*

*(After @YCor's comment) Bad wording of the consequence of Gelfand-Mazur

*(After @Yemon Choi's comments) Removed the part about $\mathbb C\{T\}$ being a PID and phrased better the question.

 A: My subjective (but, I think, not un-informed) opinion on Q1 is that one needs norm estimates if one wants to bypass Gelfand-Mazur: informally, I think of this as meaning that Fourier analysis is needed on the boundary circle, not just complex analysis on the interior of the disc. Here is one way this might be achievable — I have not worked through all the details.
I am going to denote your algebra by $W_+$ in my answer.
Let's assume that we are happy with saying that there is a natural inclusion of sets $\overline{\mathbb D} \to {\rm MaxSpec} W_+$, which sends $\lambda$ to the kernel of the evaluation map ${\rm ev}_\lambda: T\mapsto\lambda$. We would like to say that this function is surjective, i.e. that if $f\in W_+$ and ${\rm ev}_\lambda f \neq 0$ for all $\lambda\in \overline{\mathbb D}$ then $f$ lies outside every maximal ideal, equivalently $f$ is a unit in $W_+$.
As you implicitly remarked in the question and your comments, we are fine on the interior of the disc: if $f$ is holomorphic and nowhere vanishing on the open unit disc, then the same is true for $1/f$, so the only issue is whether the Taylor coefficients satisfy the "absolutely summable" condition.
Here comes a philosophical digression: if you imagine the analogous algebra $W$ in which we allow doubly infinite power series in the variable $T$, then this is the so-called Wiener algebra, and we are looking for a proof that if $f\in W$ and ${\rm ev}_\lambda f \neq 0$ for each $\lambda$ on the unit circle then $f$ is a unit in $W$. This was originally (a discrete version of) a theorem of Wiener, but the proof required Fourier-analytic estimates --- and one of the reasons why Gelfand introduced "Gelfand theory " was precisely to say that using characters and the G-M theorem one obtains a much shorter proof!
In other words, somehow your Q1 is the reverse of what Gelfand theory was invented to do.
Having said that: because there are direct proofs that a function $f$ on the unit circle which is nowhere vanishing and has absolutely summable Fourier series has the property that $1/f$ also has absolutely summable Fourier series, one should be able to take one of these proofs and apply it with the extra condition that $f$ is holomorphic on ${\mathbb D}$, to deduce that $1/f$ satisfies the right norm estimates to be in $W_+$. I have seen a fairly hands-on proof written by the non-internet-famous E. B. Davies, which used to be on his webpage before he retired; I think there is also a proof using similar ideas written by the very-internet-famous T. Tao, but I don't recall exactly which blogpost.
My thoughts on Q2 are more pessimistic, I will try to write more when I get some spare time. To get some idea why I think the closed ideal structure of $W_+$ is worse than you (probably) are hoping for, see workof Esterle--Strouse--Zouakia: Bull. Amer. Math. Soc. (1994) and J. Reine Angew. Math. (1994)
