Unification of boolean, monotone, anti-monotone, and tensor independence and L�vy processes

Abstract

The unification of the boolean and the tensor product of quantum probability spaces given by Lenczewski [Len98] is modified to include the monotone product and its mirror image, the anti-monotone product, and used to reduce the theories of boolean, monotone, and anti-monotone Levy processes on dual semi-groups to the theory of Levy processes on involutive bialgebras. This leads to a representation theorem for these processes. Further applications are a proof of the Schoenberg correspondence for boolean, monotone, and anti-monotone convolution semi-groups, a realization of boolean, monotone, and anti-monotone creation, annihilation, and conservation operators on boson Fock spaces, and a discussion of the Markov structure of these Levy processes.