Stochastic integration on the Cuntz algebra O∞

Abstract

We develop a theory of non-commutative stochastic integration with respect to the creation and annihilation process on the full Fock space over L 2( 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 L 2( 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) L 2-norm.