Deformation quantization in singular spaces

We present a method of quantizing analytic spaces X immersed in an arbitrary smooth ambient manifold M. Remarkably our approach can be applied to singular spaces. We begin by quantizing the cotangent bundle of the manifold M. Using a supermanifold framework we modify the Fedosov construction in a way such that the star-product of the functions lifted from the base manifold turns out to be the usual commutative product of smooth functions on M. This condition allows us to lift the ideals associated to the analytic spaces on the base manifold to form left (or right) ideals on (O-Omega1M[[(h) over bar]],star(h)) in a way independent of the choice of generators and leading to a finite set of PDEs defining the functions in the quantum algebra associated with X. Some examples are included. (C) 2004 American Institute of Physics.