TITLE: Fundamental group under Gelfand duality
QUESTION [11 upvotes]: Gelfand duality states that the functor of continuous functions $C(-)$ from compact Hausdorff topological to commutative $C^*$-algebras is an equivalence of categories. In other words, all topological properties of such a topological space $X$ are encoded in the algebraic properties (and of course the norm) of $C(X)$.
I have the following related questions:

How does one recognize that $X$ is simply connected from the algebraic properties of $C(X)$?
How can the fundamental group $\pi_1(X)$ be defined in terms of the algebraic properties of $C(X)$?
If $\tilde{X}$ is a universal cover of $X$, what is the algebraic relation between $C(\tilde{X})$ and $C(X)$?

REPLY [2 votes]: It seems that this question has been investigated in the literature, where the fundamental group is approached through the notion of a regular covering. For instance, the following reference establishes an equivalence between the categories of regular coverings $\tilde{X} \to X$ and Galois extensions of $C(X)$, meaning that the algebraic analog of a universal cover is a universal Galois extension and the fundamental group is the group associated to that extension:

Høst-Madsen, Anders, Separable algebras and covering spaces, Math. Scand. 87, No. 2, 211-239 (2000). ZBL1002.54010. MR1795745.

More literature on this topic is cited therein.