  We prove that every smooth diffeomorphism group valued cocycle over certain
abelian Anosov actions on tori (and more generally on infranilmanifolds), is a
smooth coboundary on a finite cover, if the cocycle is center bunched and
trivial at a fixed point. For smooth cocycles which are not trivial at a fixed
point, we have smooth reduction of cocycles to constant ones, when lifted to
the universal cover. These results on cocycle trivialisation apply, via the
existing global rigidity results, to maximal Cartan actions by Anosov
diffeomorphisms (with at least one transitive), on any compact smooth manifold.
This is the first rigidity result for cocycles over higher rank abelian
actions, with values in diffeomorphism groups, which does not require any
restrictions on the smallness of the cocycle, nor on the diffeomorphism group.
