Commutative C*-rings Let us consider the unital commutative $C^*$-algebra $C[0,1]$.  We say $A\subseteq C[0,1]$ forms a C*-subring if it satisfies the following conditions:
1- $A$ is an involutive unital  subring (closed under pointwise addtion, multiplication and invoultion  operations) whose unit is just the constant function $1$.
2- $A$ is norm-closed.
Q. Any characterization of (the elements of) $A$?
 A: This is not an answer!  It is hoped to be a reasonable conjecture, at best!
Based on Mateusz Wasilewski's comment one can build examples of subrings  of $C([0,1])$ by choosing a closed subset $K⊆[0,1]$ and taking $A$ to be the set of
functions in $C([0,1])$ taking integer values on $K$.  However this is not such a general example for at least two
reasons. It fails to include:

*

*sub-C*-algebras of $C([0, 1])$, and


*algebras of the form $R\cdot1$, where $R$ is a closed subring of $\mathbb{C}$.
Algebras of form (1) can be quite big (think of space filling curves) so the choice of $[0,1]$ as base space does not seem
to bring any relevant simplification and indeed it may actually obfuscate the problem.   So it is perhaps a good idea to
recast the problem  asking instead for the general form of subrings of $C(X)$, where $X$ is a compact topological space.
If $A$ is a subring of $C(X)$, then it is also a subring of the sub-C*-algebra generated by $A$, which in turn
is of the form $C(Y)$, where $Y$ is a quotient of $X$.  It is easy to see that $A$ separates points of $Y$ so it might
make sense to require this condition in the original question.
Regarding (2),  I haven't given much thought as to what is the general form of a closed subring of $\mathbb{C}$, such as
$\mathbb{Z}+i\mathbb{Z}$ (any ideas?), but at first glance it does not seem to be an entirely trivial question so let's do away with this
difficulty by sticking to real valued functions instead.  After all,  apart from $\mathbb{R}$ itself,  the only closed unital subring of $\mathbb{R}$ is
$\mathbb{Z}$!
So here is my conjecture:

Let $X$ be a compact Hausdorff topological space and let $A$ be a closed unital subring of $C(X, \mathbb{R})$ (continuous real valued functions on $X$), which separates points of $X$.  Then there exists a closed subset $K⊆X$
such that $A$ coincides with the set of functions in $C(X)$ taking integer values on $K$.

Hopefully a clever modification of the proof of the Stone-Weierstrass Theorem can prove this as well!
A: Let $R$ be a unital subring of $\mathbb{Z}^n$ and choose $n$ points $t_1$, $\ldots$, $t_n$ in $[0,1]$. We can see from Proposition 2.3 of this paper that $R$ can be fairly complicated. Then the set of functions in $C[0,1]$ whose restriction to $\{t_1, \ldots, t_n\}$ belongs to $R + iR$ is an involutive unital subring of $C[0,1]$. I think this shows that it is unreasonable to expect there to be any clean characterization of all involutive unital subrings of $C[0,1]$.
