Nets of Subfactors Dedicated to Bert Schroer on the Occasion of His 60th Birthday

A subtheory of a quantum eld theory speciies von Neumann subalgebras A(O) (thèob-servables' in the space-time region O) of the von Neumann algebras B(O) (thèèlds ' localized in O). Every local algebra being a (type III 1) factor , the inclusion A (O) B (O) is a sub-factor. 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 thè 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 eld net , which in turn restricts to a localized endomorphism of the observable net. The method allows to characterize , and reconstruct , local extensions B of a given theory A in terms of the observables. Various non-trivial examples are given. Several results go beyond the quantum eld theoretical application .