Paired E0-Semigroups

An E0-semigroup on a von Neumann algebra B ⊂ B(G) is a semigroup θ = {θt}t≥0 of normal unital endomorphisms θt of B. We say two E0-semigroups one, θ, on B and one, θ', on the commutant of B, B′, can be paired if there exists a normal automorphism group α on B(G) such that θ_t(b) = α_{−t}(b) θ'_t(b′) = α_t(b′) for all t ≥ 0, b ∈ B, b′ ∈ B′. We shall show: Theorem: Two strongly continuous E0-semigroups can be paired via a strongly continuous pairing α if and only if their strongly continuous product systems [Ske16] are strongly continuous commutants [Ske16] of each other. The proof goes by showing: 1. A version of the theorem without continuity: Two E0-semigroups can be paired if and only if their product systems [Ske02] are commutants [Ske03] of each other modulo a multiplier. 2. A result that may be very well more important than the other theorem itself: Theorem: The intertwiner product system `a la Arveson [Ske03] of a faithful strongly continuous E0-semigroup has sufficiently many strongly continuous sections.