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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 !

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"abelian C star algebras are the same as locally compact spaces" - no. The categories of such are (anti-)equivalent. This is not the same. –  David Roberts Dec 1 '11 at 1:33
    
Your formatting was off, so I fixed it. The question was almost incomprehensible otherwise. In response to the questions, the equivalence of categories I mentioned is one way to approach them. –  David Roberts Dec 1 '11 at 1:35
    
+1 David Roberts (first point) –  Yemon Choi Dec 1 '11 at 3:35
    
I'm sorry, but it doesn't answer at all, every grad student knows Gelfand, or simply look at Wikipedia. The problem is evidently to go beyond abelian algebras, and first to understand things on the C* algebras intrinsically, i.e. without always looking at the space. As I said, this is the starting point of K-theory, where one instead uses intuition of vector bundles. Stated in other words : what is the analog of K-theory, for discrete covers, in C-star algebra theory ? –  Amin Dec 1 '11 at 11:49
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I'm afraid that the OP is badly phrased; indeed $C_0(X)$ (the algebra of continuous functions on $X$ vanishing at infinity) is functorial only for PROPER continuous maps (we want the pullback of a $C_0$-function to be a $C_0$-function). So, for example, to describe the covering map $\mathbb{R}\rightarrow S^1$ at the $C^*$-level, you must introduce the multiplier algebra $C_b(X)$ of bounded continuous functions and (in this case) take the fixed point algebra for $\mathbb{Z}$ acting by translations, so you recover continuous periodic functions. –  Alain Valette Dec 1 '11 at 13:20

2 Answers 2

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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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 http://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.

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It's a correct statement, but I remain to be convinced that the forced unitization aka the one-point compactification is the right way to think about these things. Most C-star algebraists find the multiplier algebra more natural in many cases. See for instance all the stuff on C-star algebraic quantum groups –  Yemon Choi Sep 15 '12 at 18:29
    
(I learned all this from comments by Matthew Daws, though I don't have a specific link at hand) –  Yemon Choi Sep 15 '12 at 18:30
    
Let $X$ and $Y$ be locally compact Hausdorff spaces. If I recall correctly, $C_0(\cdot)$ is contravariantly functorial for proper maps from an open subset $U \subseteq X$ to $Y$. The induced map is the composition of the pullback $C_0(Y) \to C_0(U)$ and the inclusion $C_0(U) \to C_0(X)$, and every $\ast$-homomorphism $C_0(Y) \to C_0(X)$ is induced by one of these maps. If I'm remembering all this correctly, LCH spaces with these sorts of maps is the right category for expressing Gelfand duality for non-unital commutative C*-algebras. –  Paul Siegel Sep 15 '12 at 22:06
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@Paul: yes, I think that specifying such a map is equivalent to specifying a continuous pointed map between the one-point compactifications of $X$ and $Y$. –  Qiaochu Yuan Sep 15 '12 at 22:33

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