What type of dynamics arise in E_0-dilations of commuting quantum Markov process?

Abstract

Let H be a separable Hilbert space. Given two strongly commuting CP_0-semigroups $\phi$ and $\theta$ on B(H), there is a Hilbert space K containing H and two (strongly) commuting E_0-semigroups $\alpha$ and $\beta$ such that $\phi_s \circ \theta_t (P_H A P_H) = P_H \alpha_s \circ \beta_t (A) P_H$ for all s,t and all A in B(K). In this note we prove that if $\phi$ is not an automorphism semigroup then $\alpha$ is cocycle conjugate to the minimal *-endomorphic dilation of $\phi$, and that if $\phi$ is an automorphism semigroup then $\alpha$ is also an automorphism semigroup. In particular, we conclude that if $\phi$ is not an automorphism semigroup and has a bounded generator (in particular, if H is finite dimensional) then $\alpha$ is a type I E_0-semigroup.