Twisted Lie group $C^*$-algebras as strict quantizations

A nonzero 2-cocycle $\Gamma\in Z^2(\g,\R)$ on the Lie algebra $\g$ of a compact Lie group $G$ defines a twisted version of the Lie-Poisson structure on the dual Lie algebra $\g^*$, leading to a Poisson algebra $C^{\infty}(\g_{(\Gamma)}^*)$. Similarly, a multiplier $c\in Z^2(G,U(1))$ on $G$ which is smooth near the identity defines a twist in the convolution product on $G$, encoded by the twisted group $C^*$-algebra C^*(G,c)$. Further to some superficial yet enlightening analogies between$C^{\infty}(\g^*_{(\Gamma)})$and$C^*(G,c)$, it is shown that the latter is a strict quantization of the former, where Planck's constant$\hbar$assumes values in$(\Z\backslash\{0\})^{-1}$. This means that there exists a continuous field of$C^*$-algebras, indexed by$\hbar\in 0\cup (\Z\backslash\{0\})^{-1}$, for which$\A^0=C_0(\g^*)$and$\A_{\hbar}=C^*(G,c)$for$\hbar\neq 0$, along with a cross-section of the field satisfying Dirac's condition asymptotically relating the commutator in$\A_{\hbar}$to the Poisson bracket on$C^{\infty}(\g^*_{(\Gamma)})$. Note that the `quantization' of$\hbar$does not occur for$\Gamma=0$.