$\newcommand{\g}{\mathfrak g}\newcommand{\h}{\hbar}$ For a finite dimensional Lie algebra $\g$, he Duflo isomorphism is a complicated algebra isomorphism between the $\g$-invariant part $S(\g)^\g$ of the symmetric algebra, and the center $Z(U(\g)$ of the enveloping algebra.

One way to undertsand this result is as follow : $S=S(\g)$ is identified with the algebra of polyonmial functions on $\g^*$. The Lie bracket of $\g$ turns this algebra into a Poisson algebra. Let $U_{\h}$ be the enveloping algebra of $\g_{\h}$, which is $\g[[\h]]$ in which the bracket is multiplied by $\h$. In particular $U_0=S$ and the PBW theorem implies that $U_\h$ is a flat deformation of $S$. Hence there is a linear (in fact $\g$-linear) isomorphism $S[[\h]]\rightarrow U_\h$. The pull-back of the product in $U_\h$ through this isomorphism induces a star product $m_{pbw}$ on $S[[\h]]$ which is more or less by definition a quantization of the Poisson algebra $S(\g)$.

The space $S(\g)^\g$ is the Poisson center of $S(\g)$. Then Duflo theorem can be rephrased by saying that there exists a star-product $m_d$ whose restriction to the Poisson center is undeformed, i.e. identical to the original product on $S(\g)^g$.

Two star-products are said to be equivalent if they are related by an algebra isomorphism which is the identity modulo $\h$. Equivalence classes of star product are essentially classified by the Hochschild cohomology.

It turns out that while $m_d$ satisfies a strong property which $m_{pbw}$ doesn"t, they are equivalent as star products. Hence my question is:

Is there a finer equivalence relation on star-products which distinguishes between those ?

Or perhaps

Is there an appropriate replacemnt for Hochschild cohomology which classify star-products on Poisson algebras respecting the Poisson center ?

A sort of obvious guess would be cyclic cohomology since both cyclic cohomology and the Duflo star-product have very much to do with traces. Yet it seems that cyclic cohomology is relvant only for unimodular Poisson manifolds, and describes so-caled closed star-product respecting a specific trace rather than the space of those.

**edit**: I found some answers here. They basically give a sufficient condition for a star product on $A$ to be trivial on $C$ in term of the kernel of the restriction map
$$CH(A,A)\rightarrow CH(C,A).$$

They also claim without full details that under mild assumption, if there is a star product on $A$ whose restriction to $C$ is commutative then this star product is equivalent to one whose restriction to $C$ is actually trivial. This sound surprising since if it were true in the case I'm interested in it would give a rather easy proof of Duflo's theorem...