Non-commutative analysis on quantum spaces

The present thesis deals with the transfer of elements of the non-commutative analysis to the commutative space. First a possibility is presented, how the product of two elements of the non-commutative algebra can be transferred to the commutative algebra. By the application of special vector fields a generalized star product is given in the form of a closed formula, so that perturbation-theoretical approaches can be generalized. The resulting action is explicitly represented for the case of a q-deformed Euclidean space in n dimensions as well as the Leibniz rule changed by the non-commutativity. The lattice structure of the space induced by partial differentiations is used in order to express the determined integral as sum over the function values at all lattice points. The constructed Hilbert space yields the necessary mathematical basis and the possibility to represent the defined integral as trace of a special trace-class operator