Quantum Poisson processes and dilations of dynamical semigroups

Abstract

The notion of a quantum Poisson process over a quantum measure space is introduced. This process is used to construct new quantum Markov processes on the matrix algebraM n with stationary faithful state π. If (ℳ, μ) is the quantum measure space in question (ℳ a von Neumann algebra and μ a faithful normal weight), then the semigroupe tL of transition operators on (M n , π) has generator whereu is an arbitrary unitary element of the centraliser of (M n ⊗ℳ,φ⊗μ).