Star Products with Separation of Variables Admitting a Smooth Extension

Given a complex manifold M with an open dense subset ?? endowed with a pseudo-K?hler form ?? which cannot be smoothly extended to a larger open subset, we consider various examples where the corresponding K?hler?CPoisson structure and a star product with separation of variables on (??, ??) admit smooth extensions to M. We give a simple criterion of the existence of a smooth extension of a star product and apply it to these examples.