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

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

2007-11-19

arxiv.org/abs/0711.2876v2

Infin. Dimens. Anal. Quantum Probab. Relat. Top. 11/3 (2008), 393-403

Mathematics

Operator Algebras

9 pages, minor corrections made

Scientific paper

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.

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