On the product system of a completely positive semigroup

Representations of product systems over semigroups and dilations of commuting CP maps

“…M by Θ(a) = T (I E ⊗ a) T * and Φ(a) = S(I F ⊗ a) S * for a ∈ M. Let X Θ,Φ be the product system constructed in the proof of "(1) implies (2)" of Proposition 5.6 (so that it is defined by (E Θ , E Φ , t Θ,Φ ) and t Θ,Φ is as in (18)) and let (id, T Θ , T Φ ) be the identity representation of X Θ,Φ .…”

“…When the semigroup is not discrete, one usually assumes certain continuity or measurability conditions on the product system. Product systems of C * -correspondences over R + or subsemigroups of R + were studied by various authors (e.g., [5,8,12,18,21,31] and others). Of course, a single correspondence can also be thought of as a product system over the semigroup N.…”

Quantized convolution semigroups

“…To complete the program, it would be necessary to prove that the induced weak isomorphism is in fact measurable and hence a bona fide isomorphism. This technical fact has been in fact accomplished in [Mar02b].…”

“…We proceed to review succintly our construction given in full detail in [Mar02b]. Given a CP semigroup (φ t ) t≥0 of B(H 0 ) with minimal dilation α, we obtain a product system E φ that is canonically isomorphic to E α .…”

Quantum Markov Semigroups: Product Systems and Subordination

“…As we showed in [26], it is possible to "refine" the family {E t } t≥0 in order to obtain a product system {E Θ (t)} t≥0 over N ′ . This process was also carried out in [22] and a "dual" process was used in [10]. To describe the process from [26] briefly, we fix t > 0 and for any partition P = {t 0 = 0 < t 1 < .…”