# Nets of Subfactors

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.

Keywords:

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