Stochastic integration on the Cuntz algebra O∞

AbstractWe develop a theory of non-commutative stochastic integration with respect to the creation and annihilation process on the full Fock space over L2(R). Our theory largely parallels the theories of non-commutative stochastic Itô integration on Boson and Fermion Fock space as developed by R. Hudson and K. R. Parthasarathy. It provides the first example of a non-commutative stochastic calculus which does not depend on the quantum mechanical commutation or anticommutation relations, but it is based on the theory of reduced free products of C∗-algebras by D. Voiculescu. This theory shows that the creation and annihilation processes on the full Fock space over L2(R), which generate the Cuntz algebra O∞, can be interpreted as a generalized Brownian motion. We should stress the fact that in contrast to the other theories of stochastic integration our integrals converge in the C∗-norm on O∞, i.e., uniformly rather than in some state-dependent strong operator topology or (non-commutative) L2-norm