Schoenberg Correspondence on Dual Groups
  As in the classical case of L\'evy processes on a group, L\'evy processes on a Voiculescu dual group are constructed from conditionally positive functionals. It is essential for this construction that Schoenberg correspondence holds for dual groups: The exponential of a conditionally positive functional is a convolution semigroup of states. 
Michael Sch\"urmann and Stefan Vo\ss
math.OA math.PR