mathoverflow.net/questions/372841/…

@DavidRoberts I am reading your post below to have a better imagination of the story: I think you are indicating to woronowich morphism. any way what would be happen if I simply assume that the morphisms are usual morphisms between C^* algebras(According to my main question in this post)?

Do you mean that $C_0(X,A\oplus A)\sim C_0 (X\times X,A)$ hence isomorphic to $C_0(X,A)$ since $X$ homeomorphic to $X\times X$? Or you mean some things else?

@Alejandro The symplectic version of centralizer problem is also a related question. Plz see the last lines of this post mathoverflow.net/questions/193650/…

@DavidRoberts Yes w consider all possible morphisms(unital or non unital). However the "exclusively unital morphisms" produce another question

@BenMcKay I do not know what is Palais theorem precisely but it seems that your comment somewhat is also applicable to Palais theorem too(according to $m\circ \nabla=\lambda d$ for a constant $\lambda$

what is Palais theorem precisely?

do you mean $m: \Lambda^1 \otimes \Lambda^1 \to \Lambda^2$?

@kvicente but as you interestingly indicated invariant finit measure subsets of the cotangent bundle is a necessary point which can generates new interesting question

@kvicente however i wonder if ergodicity can be defined in non finite measure too: a full measure set is a measurable set whose intersection with every compact set K has full measure $\mu(K)$

@kvicente what about isometric condition? namely we assume that f preserves a Riemannian metric However the volum form of the cotangent bundle is independent of any metric on M

@kvicente very interesting and necessary point. thank you I come back very soon

@DavidRoberts May be I did not understand well your previous comment?

mathoverflow.net/questions/451247/…

Any way your question has a beautiful philosophy when certain property of f on M carries to the same property for f^* on the cotangent bundle. on can ask some questions of this type in for example Ergodic theory. Assume f is ergodic is the pull back an ergodic map too? please see the following:

In the other word what is a map (time one map of flow) on the plane which does not come from a circle action?