Deformation quantization of commutative families

In my talk I will describe a series of cohomological obstructions, that regulate the existence of elements in deformation quantization of a Poisson manifold, that would extend the given commutative (w.r. to the Poisson brackets) family of smooth functions to a family of commuting (in the usual sense) elements in the deformed algebra of functions on this manifo ld. In the end I will describe a conjecture, that gives a more geometric way to construct such obstructions in the particular case, when the manifold in question is symplectic. The talk is based on work, supported by the grant NSh-6399.2018.1.