Paired E0–semigroups

In these notes we prove two main results: 1) It is well-known that two strongly continuous E_0-semigroups on can be paired if and only if they have anti-isomorphic Arveson systems. For a new notion of pairing (which coincides only for B(H) with the existing one), we show: For a von Neumann algebra B, a strongly continuous E_0-semigroup theta on B and a strongly continuous E_0-semigroup theta' on B' can be paired if and only if their product systems are commutants of each other. 2) On the way to prove the former, en passant we have to fill in a long standing important gap in the theory of intertwiner product systems à la Arveson (known, so far, only for B(H) in the separable case): Intertwiner product systems of faithful strongly continuous E_0-semigroups on von Neumann algebras have sufficiently many strongly continuous sections. We explain why both results are entirely out of reach for Arveson's methods [Arv89,Arv90] and depend essentially on the alternative approach from Skeide [Ske16].