Star Product on the Euclidean Motion Group in the Plane

In this work, we perform exact and concrete computations of star-product of functions on the Euclidean motion group in the plane, and list its $C$-star-algebra properties. The star-product of phase space functions is one of the main ingredients in phase space quantum mechanics, which includes Weyl quantization and the Wigner transform, and their generalizations. These methods have also found extensive use in signal and image analysis. Thus, the computations we provide here should prove very useful for phase space models where the Euclidean motion groups play the crucial role, for instance, in quantum optics.