A Counterexample to the Quantizability of Modules

Let π be a Poisson structure on Rn vanishing at 0. It leads to a Kontsevich type star product ⋆π on C∞(Rn)[[ǫ]]. We show that 1. The evaluation map at 0 ev0 : C ∞(Rn) → C can in general not be quantized to a character of (C∞(Rn)[[ǫ]], ⋆π). 2. A given Poisson structure π vanishing at zero can in general not be extended to a formal Poisson structure πǫ also vanishing at zero, such that ev0 can be quantized to a character of (C∞(Rn)[[ǫ]], ⋆πǫ). We do not know whether the second claim remains true if one allows the higher order terms in ǫ to attain nonzero values at zero. How to read this paper in 2 minutes The busy reader can take the following shortcut: 1. Read Theorem 6 on page 3 for the main result. 2. Read Definition 2 if its statement is not clear. 3. Look at eqns. (13) and the preceding enumeration for the definition of the counterexample.