On the generators of quantum stochastic flows

Abstract

A time-indexed family of *-homomorphisms between operator algebras (j_t:A→B)t∈I is called a stochastic process in quantum probability. When Ε C:B →C is a conditional expectation onto a subalgebra, the composed process (k_t=C ^° j_t)t∈I is no longer *-homomorphic, but is completely positive and contractive. In some situations, the filtered process k may be described by a stochastic differential equation. The central aim of this paper is to study completely positive processes k which admit a differential description through a stochastic equation of the formed k_t=k_t°θ^α_β Λ^β_α(t), in which Λ is the matrix of basic integrators of finite dimensional quantum stochastic calculus, and θ is a matrix of bounded linear maps on the algebra. The structure required of the matrix θ, 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 equation dV_t=l^α_βV_t dΛ^β_α(t) in which l is a matrix of bounded Hilbert space operators.