OPERATOR SPACE STRUCTURES and the SPLIT PROPERTY II The canonical non-commutative L embedding

A characterization of the split property for an inclusion N ⊂ M of W ∗-factors with separable predual is established in terms of the canonical non-commutative L embedding considered in [4, 5] Φ2 : a ∈ N → ∆ M,ΩaΩ ∈ L(M,Ω) associated with an arbitrary fixed standard vector Ω for M . This characterization follows an analogous characterization related to the canonical non-commutative L embedding Φ1 : a ∈ N → (·Ω, JM,ΩaΩ) ∈ L(M,Ω) also considered in [4, 5] and studied in [20]. The split property for a Quantum Field Theory is characterized by equivalent conditions relative to the non-commutative embeddings Φi, i = 1, 2, constructed by the modular Hamiltonian of a privileged faithful state such as e.g. the vacuum state. The above characterization would be also useful for theories on a curved space-time where there exists no a-priori privileged state. Mathematics Subject Classification Numbers: Primary 46L35, 47D15, Secondary 47D25.