Quantum stochastic processes for maps on Hilbert C*-modules

Abstract

We discuss pairs (ϕ, Φ) of maps, where ϕ is a map between C*-algebras and Φ is a ϕ-module map between Hilbert C*-modules, which are generalization of representations of Hilbert C*-modules. A covariant version of Stinespring's theorem for such a pair (ϕ, Φ) is established, and quantum stochastic processes constructed from pairs ({ϕt}, {Φt}) of families of such maps are studied. We prove that the quantum stochastic process J = {Jt} constructed from a ϕ-quantum dynamical semigroup Φ = {Φt} is a j-map for the quantum stochastic process j = {jt} constructed from the given quantum dynamical semigroup ϕ = {ϕt}, and that J is covariant if the ϕ-quantum dynamical semigroup Φ is covariant.