Operator space structures and the split property II

TL;DR

This paper characterizes the split property for inclusions of $W^*$-factors using non-commutative $L^2$ and $L^1$ embeddings, providing a framework applicable even without a privileged state, relevant for quantum field theories on curved spacetime.

Contribution

It introduces a new characterization of the split property via non-commutative embeddings, extending previous work and applicable to curved spacetime quantum field theories.

Findings

Characterization of the split property using $L^2$ embedding.

Equivalent conditions for the split property via $L^1$ embedding.

Applicability to quantum field theories on curved spacetime.

Abstract

A characterization of the split property for an inclusion $N\subset M$ 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.