On the Generators of Quantum Stochastic Flows

A time-indexed family of ∗-homomorphisms between operator algebras (jt: A→B)t∈Iis called a stochastic process in quantum probability. When EC: B→C is a conditional expectation onto a subalgebra, the composed process (kt=EC∘jt)t∈Iis no longer ∗-homomorphic, but is completely positive and contractive. In some situations, the filtered processkmay be described by a stochastic differential equation. The central aim of this paper is to study completely positive processeskwhich admit a differential description through a stochastic equation of the formdkt=kt∘θαβ Λβα(t), in whichΛis the matrix of basic integrators of finite dimensional quantum stochastic calculus, andθis a matrix of bounded linear maps on the algebra. The structure required of the matrixθ, for complete positivity of the process, is obtained. The stochastic generators of contractive, unital, and ∗-homomorphic processes are also studied. These results are applied to the equationdVt=lαβVt dΛβα(t) in whichlis a matrix of bounded Hilbert space operators.