What type of dynamics arise in E_0-dilations of commuting quantum Markov process?
Abstract
Description
Let H be a separable Hilbert space. Given two strongly commuting CP_0-semigroups $ϕ$ and $θ$ on B(H), there is a Hilbert space K containing H and two (strongly) commuting E_0-semigroups $α$ and $β$ such that $ϕ_s \circ θ_t (P_H A P_H) = P_H α_s \circ β_t (A) P_H$ for all s,t and all A in B(K).
In this note we prove that if $ϕ$ is not an automorphism semigroup then $α$ is cocycle conjugate to the minimal *-endomorphic dilation of $ϕ$, and that if $ϕ$ is an automorphism semigroup then $α$ is also an automorphism semigroup. In particular, we conclude that if $ϕ$ is not an automorphism semigroup and has a bounded generator (in particular, if H is finite dimensional) then $α$ is a type I E_0-semigroup.
9 pages, minor corrections made
9 pages, minor corrections made