Concerning covering spaces, there is a paper by Evgenij Troitsky and Alexander Pavlov titled Quantization of branched coverings. In particular, they have the following theorem.

Suppose $i \colon C(X) \to C(Y)$ is an
inclusion, where $X$ and $Y$ are
compact Hausdorff spaces. Let $p = i^*
> \colon Y \to X$ be the projection
which is Gelfand dual to $i$. Then the
following are equivalent:

(a) $p$ is a branched covering (i.e.
it is a closed and open continuous
surjection with a finite bounded
number of preimages #$p^{-1}(x)$).

(b) There exists a positive unital
conditional expectation $E \colon C(Y)
> \to C(X)$, which is topologically of finite
index.

The notion of a branched covering is of course weaker than that of a finite covering. Regarding the latter, you will find the following theorem in the paper cited above:

Suppose $i \colon C(X) \to C(Y)$ is an
inclusion, where $X$ and $Y$ are
compact Hausdorff spaces. Let $p = i^*
> \colon Y \to X$ be the projection
which is Gelfand dual to $i$. Then the
following are equivalent:

(a) $p$ is a finite covering.

(b) There exists a positive unital
conditional expectation $E \colon C(Y)
> \to C(X)$, which is algebraically of finite
index.

(c) The module $C(Y)$ may be equipped with a $C(X)$-valued inner product in such a way that it becomes a finitely generated projective Hilbert $C(Y)$-module.

You might wonder about the definition of *topologically of finite index* and *algebraically of finite index* in the statements above. The definitions are as follows:

Given a $C^*$-algebra $B$ and a $C^*$-subalgebra $A \subset B$. A conditional expectation $E \colon B \to A$ is *topologically of finite index* if the mapping $(C \cdot E - id_B)$ is positive for some real number $K \geq 1$.

... and ...

Given a $C^*$-algebra $B$ and a $C^*$-subalgebra $A \subset B$. A conditional expectation $E \colon B \to A$ is *algebraically of finite index* if there exists a family $\{u_1, \dots, u_n\} \subset B$, such that
$$
b = \sum_{i=1}^n u_i E(u_i^*b)
$$
The set $\{u_1, \dots, u_n\}$ is called a quasi-basis of $E$.

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