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:

A closed manifold with a subset with the same ring cohomology


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