Deformation quantization on the cotangent bundle of a Lie group

Abstract

We develop a complete theory of non-formal deformation quantization on the cotangent bundle of a weakly exponential Lie group. An appropriate integral formula for the star-product is introduced together with a suitable space of functions on which the star-product is well defined. This space of functions becomes a Fréchet algebra as well as a pre-C * -algebra. Basic properties of the star-product are proved and the extension of the star-product to a Hilbert algebra and an algebra of distributions is given. A C * -algebra of observables and a space of states are constructed. Moreover, an operator representation in position space is presented. Finally, examples of weakly exponential Lie groups are given.