3 Review(s)

The algebras discussed here have many common features with the algebra of complete symbols of pseudodifferential operators, except that in general there are no corresponding operator algebras. Nevertheless, the developed calculus allows definition of the notion of an elliptic element and its index as well as proving an index theorem similar to that of Atiyah-Singer for elliptic operators. The corresponding index formula contains the Weyl curvature and the usual ingredients entering the Atiyah-Singer formula. Applications of the index theorem are connected with the so-called asymptotic operator representation of the deformed algebra (the operator quantization), the formal deformation parameter 'h' should be replaced by a numerical one ranging over some admissable set of the unit interval having 0 as its limit point. The fact that the index of any elliptic operator is an integer results in necessary quantization conditions: the index of any elliptic element should be asymptotically integer-valued as 'h' tends to 0 over the admissible set. Finally, a reduction theorem for deformation quantization is proved generalizing the classical Marsden-Weinstein theorem.