Commutative spectral triples not coming from manifolds

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 If the commutative spectral triple does not obey these axioms the theorem is no longer valid: I've heard that there are some nice examples of commutative spectral triples which in fact don't come from manifolds (I asked also a somehow related question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.

• You might have a look at this paper arxiv.org/abs/1010.0156 which constructs spectral triples for $C(X)$ where $X$ is a compact metric space. Their particular examples come from tiling spaces and dynamics, where $X$ is a Cantor set. – mkreisel Dec 29 '14 at 23:14
• In particular, you might also take a look at this paper, arxiv.org/abs/1112.6401, where the authors construct a family of spectral triples for the Sierpinski gasket, which, for instance, can be made to have spectral dimension the Hausdorff dimension of the Sierpinski gasket. – Branimir Ćaćić Dec 30 '14 at 0:24
• Another very interesting example, coming from a purely differential-geometric context, is Hasselmann's recent work on constructing spectral triples for Carnot manifolds: arxiv.org/abs/1404.5494. – Branimir Ćaćić Dec 30 '14 at 23:17