I'm interested in taking the noncommutative quotient of a manifold, and endowing it with a kind of noncommutative smooth structure. More formally I'm interested in the question: is there a canonical way to construct a Spectral Triple on a crossed product algebra(=noncommutative quotient).
What do I mean with canonical? Well I would hope that at the very least it should return the quotient manifold in the commutative case.
Let $M$ be a spin manifold, $G$ a group acting on $M$ such that $M/G$ is a manifold, let $(C(M),H_1,D_1)$ be the spectral triple of $M$, $(C(M/G),H_2,D_2)$ the spectral triple of $M/G$, and let $A=C(M) \ltimes G$ the noncommutative quotient. Then I would want a suitable spectral triple $(A',H_3,D_3)$ on $A$ such that it is Morita equivalent to $(C(M/G),H_2,D_2)$.
Moreover I would like it to be Morita equivalent via the Morita equivalence of $C(M/G)$ and $A$. I should note that for two spectral triple to be Morita equivalent is somewhat tricky, see Varilly's "Introduction to Noncommutative Geometry" Chapter 7.
In the most ideal case it would also reproduce the spectral triple structure in cases we already have a natural construction, for instance the spectral triples on the noncommutative tori.