Question

Compact Hausdorff spaces bijectively correspond to C^*-algebras with identity. One needs to consider the algebra of continuous functions C(X) to go in one direction and spectrum to go in the other. (See e.g. Wikipedia). The situation is similar to algebraic geometry - affine manifolds correspond to commutative algebras... Basic skill in alg.geom. is to recast algebraic properties in geometric and vice versa e.g. projective modules - vector bundles... (the dictionary is lengthy).

I wonder about similar correspondence in C^*-algebra setup. In particular:

Question 1: if space "X" is topological manifold (i.e. locally R^n), is there some "nice" way to recognize it via C^*-algebra of continuous function ? (... is there non-commutative version ? ... )

Question 2: if "X" is smooth manifold, is there nice way to recognize it and define sub-algebra of smooth functions entirely in terms of C^*-algebra ? (... is there non-commutative version ? ... )

Question 3 Is it possible to characterize the set of all measures on "X" in terms of C(X) ? (... is there non-commutative version ? ... )

If you have further comments how interesting algebraic properties can be recasted in topological or vice versa, you are welcome to post.

Answer 1

Question 1: The topological $n$-manifold property is equivalent to every point of $X$ having a neighborhood homeomorphic to $B^n$, the closed unit ball in $\mathbb{R}^n$. The existence of such a neighborhood $x\in B^n_x \subset X$ induces the surjective algebra homomorphisms $C(X) \to C(B_x^n) \cong C(B^n) \to C(\{x\})\cong \mathbb{R}$ (actually, extremal epimorphisms, I think). The Gelfand duality between compact Hausdorff topological spaces and commutative real $C^*$ algebras ensures that the existence of surjective homomorphisms $C(X) \to C(B^n) \to \mathbb{R}$ (the first map should be an extremal epimorphism, while the second should correspond to the quotient by the maximal ideal of an interior point of $B^n$) implies the existence of continuous maps $\{x\} \to B^n \to X$, where $x$ maps to an interior point of $B^n$ and $B^n$ is embedded in $X$, and hence a neighborhood of $x$ in $X$. Having such such algebra homomorphisms for each $x\in X$ characterizes $X$ as a topological manifold.

Question 2: $C^\infty(X)$ for a compact manifold $X$ is not a $C^*$ algebra. It is at the very least a Fréchet algebra, where multiplication satisfies an extra convexity condition (though I'm fuzzy on the details). It is at least clear that one must leave the category of $C^*$ to characterize it. A point that non-commutative geometry centered discussions of this questions seem to be ignoring is that there already exists an algebraic characterization of $C^\infty(X)$ that has nothing to do with non-commutative geometric spectral triples. In my understanding, such a characterization can be found in an article by Michor and Vanžura (arXiv:math/9404228).

Question 3: As already mentioned in Vahid's answer, this is the content of the Riesz representation theorem. The topological vector space dual to $C(X)$ is the space of signed Radon measures on $X$. $C(X)$ is partially ordered by pointwise comparison, which also induces a partial order on its dual. The positive cone in the space of signed Radon measures consists of the positive Radon measures.

I cannot say anything about non-commutative versions of the above answers. But, since these correspondences are heavily based on lots of non-trivial maximal ideals, and such ideals are likely to be absent in non-commutative algebras, they probably do not translate directly.

Answer 2

The answer to your first (and second) question is negative, because commutative C*-algebras only reflect global features of the underlying space.
For the third question: we have the Riesz representation theorem which says: For locally compact and Hausdorff topological space $X$, there is and isometric isomorphism between the dual of $C_0(X)$ and the space of radon measures on $X$. See Theorem 7.17 in Folland's book "Real analysis".