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 $\varphi$ 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 ${\varphi }_s\circ {\theta }_t(P_{H}A{P}_H)=P_H{\alpha }_s\circ {\beta }_t(A)P_H$ for all s,t and all A in B(K). In this note we prove that if $\varphi$ is not an automorphism semigroup then $\alpha$ is cocycle conjugate to the minimal *-endomorphic dilation of $\varphi$, and that if $\varphi$ is an automorphism semigroup then $\alpha$ is also an automorphism semigroup. In particular, we conclude that if $\varphi$ 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.

9 pages, minor corrections made

References in corpus (1)

• E_0-dilation of strongly commuting CP_0-semigroups