There is a very deep and remarkable theorem by Connes (the so called reconstruction theorem) which states that from a commutative spectral triple obeying certain axioms one can reconstruct a smooth compact oriented manifold. If the commutative spectral triple does not obey these axioms the theorem is no longer valid: I've heard that there are some nice examples of commutative spectral triples which in fact don't come from manifolds (I asked also a somehow related question in this topic: Commutative spectral triples). I'm interested in knowing some nice examples of such spectral triples: I'm also interested in several questions concerning the structure of such spectral triples:
Are all of them of the form $(A,H,D)$ where $A \subset C(X)$ for some compact topological space $X$? Are there some examples of this form with $X$-infinite dimensional?
If $A$ is a subalgebra of $C(X)$ what is the relation between the dimension spectrum of $(A,H,D)$ and the usual dimension of $X$?
Please forgive me if this question is too broad: I will then try to separate it into more than one.