Abstract
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
2
embedding considered in \cite{B1,B2}
\F_2:a\in N\to
\D_{M,\Om}^{1/4}a\Om\in L^2(M,\Om)
associated with an arbitrary fixed standard vector $\Om$ for
M
. This characterization follows an analogous characterization related to the canonical non-commutative
L
1
embedding
\F_1:a\in N\to (\cdot\Om,J_{M,\Om}a\Om)\in L^1(M,\Om)
also considered in \cite{B1,B2} and studied in \cite{F}. The split property for a Quantum Field Theory is characterized by equivalent conditions relative to the non-commutative embeddings $\F_i$,
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.