An Isomorphism of Quantum Semimartingale Algebras

We give details of a *‐linear bijection between adapted (in the sense of Hudson and Parthasarathy) and vacuum‐adapted quantum stochastic integrals. This provides new insight into Attal's remarkable transformation of quantum semimartingales, by showing that it factorizes in a natural manner. The Banach *‐algebras of regular quantum and Ω‐semimartingales are consequently isomorphic, and an intrinsic characterisation of Ω‐semimartingales is obtained as an application of this fact. Various formulae occurring in quantum stochastic calculus are shown to have a more natural appearance in the vacuum‐adapted framework. We finish by providing the full generalization of this theory to higher dimensions.