Star product realizations of $\kappa$-Minkowski space

Abstract

We define a family of star products and involutions associated with $\kappa$-Minkowski space. Applying corresponding quantization maps we show that these star products restricted to a certain space of Schwartz functions have isomorphic Banach algebra completions. For two particular star products it is demonstrated that they can be extended to a class of polynomially bounded smooth functions allowing a realization of the full Hopf algebra structure on $\kappa$-Minkowski space. Furthermore, we give an explicit realization of the action of the $\kappa$-Poincaré algebra as an involutive Hopf algebra on this representation of $\kappa$-Minkowski space and initiate a study of its properties.