Galois cover via C star algebras Hello to all, here's my question, I hope it's not too trivial. I haven't found reference for it so far.
We know that abelian C star algebras are the same as locally compact spaces. 
Now what is the framework in the realm of C*-algebras to describe :


*

*spaces that are locally homeomorphic ?

*Galois covers (=finite coverings) of a given space ? 


I suspect that for the second question, one could try to take quotients of finite copies of the C*-algebra of the base, and probably mimic what is done with projectors for vector bundles, but again, for such a natural problem, I haven't found reference.
Thanks !
 A: 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 $E$.

A: This is not an answer to the question, but regarding the discussion in the comments, commutative C*-algebras are not antiequivalent to locally compact Hausdorff spaces even if one restricts attention to proper maps. See https://math.stackexchange.com/questions/170984/are-commutative-c-algebras-really-dual-to-locally-compact-hausdorff-spaces . The correct statement is that commutative C*-algebras are antiequivalent to pointed compact Hausdorff spaces. 
