Quantum Bernoulli experiments and quantum stochastic processes

Abstract

Based on a W*-algcbraic approach to quantum probability theory we construct basic discrete internal quantum stochastic processes with independent increments. We obtain a one-parameter family of (classical) Bernoulli experiments as linear combinations of these basic processes.

Then we use the nonstandard hull of the internal GNS-Hilbert space $\mathcal{H}_\tau$ corresponding to the chosen state τ (the underlying quantum probability measure) in order to derive nonstandard hulls of our internal processes. Finally continuity requirements lead to the specification of a certain subspace $\mathcal{L}$ of to which the nonstandard hulls of our internal processes can be restricted and which turns out to be isomorphic to the Loeb-Guichardet space introduced by Leitz-Martini [10]. A subspace of $\mathcal{L}$ then is shown to be isomorphic to the symmetric Fock space $\mathcal{F}_ +$(L ²([0,1], λ)) and our basic processes agree with the processes of Hudson and Parthasarathy on this subspace.