On the Algebraic Foundations of Non-Commutative Probability Theory

As a first part of a rigorous mathematical theory of non-commutative probability we present, starting from a set of canonical axioms, a complete classification of the notions of non-commutative stochastic independence. Our result originates from a first contribution and a conjecture by M. Schurmann [21; 22] and is based on a fundamental paper by R. Speicher [26]. Our concept is used to initiate a theory of non-commutative L'evy processes which are defined on dual groups in the sense of D. Voiculescu [29]. The paper generalises L'evy processes on Hopf algebras [20] to non-commutative independences other than the `tensor' independence of R. L. Hudson [9, 12]