# Nets of Subfactors

Roberto Longo, Karl-Henning Rehren
November 10, 1994
A subtheory of a quantum field theory specifies von Neumann subalgebras ${\cal A(O)}$ (the observables' in the space-time region ${\cal O}$) of the von Neumann algebras ${\cal B(O)}$ (the fields' localized in ${\cal O}$). Every local algebra being a (type III$_1$) factor, the inclusion ${\cal A(O)} \subset {\cal B(O)}$ is a subfactor. The assignment of these local subfactors to the space-time regions is called a net of subfactors'. The theory of subfactors is applied to such nets. In order to characterize the relative position' of the subtheory, and in particular to control the restriction and induction of superselection sectors, the canonical endomorphism is studied. The crucial observation is this: the canonical endomorphism of a local subfactor extends to an endomorphism of the field net, which in turn restricts to a localized endomorphism of the observable net. The method allows to characterize, and reconstruct, local extensions ${\cal B}$ of a given theory ${\cal A}$ in terms of the observables. Various non-trivial examples are given.

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