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.