What is a relevant  non commutative analogues for the following fact, in term of spectral triples and cyclic cohomology?:

"If $M$ is a compact oriantable manifold without boundary and $X\subset M$ is a proper subset
with inclusion $i: X \to M$, then $i^*$ is not a ring isomorphism in cohomology"

I asked the commutative version in the following post:

https://mathoverflow.net/questions/156228/a-closed-manifold-with-a-subset-with-the-same-ring-cohomology