  Starting from a local quantum field theory with an unbroken compact symmetry
group $G$ in 1+1-dimensional spacetime we construct disorder fields
implementing gauge transformations on the fields (order variables) localized in
a wedge region. Enlarging the local algebras by these disorder fields we obtain
a nonlocal field theory, the fixpoint algebras of which under the appropriately
extended action of the group $G$ are shown to satisfy Haag duality in every
simple sector. The specifically 1+1 dimensional phenomenon of violation of Haag
duality of fixpoint nets is thereby clarified. In the case of a finite group
$G$ the extended theory is acted upon in a completely canonical way by the
quantum double $D(G)$ and satisfies R-matrix commutation relations as well as a
Verlinde algebra. Furthermore, our methods are suitable for a concise and
transparent approach to bosonization. The main technical ingredient is a
strengthened version of the split property which should hold in all reasonable
massive theories. In the appendices (part of) the results are extended to
arbitrary locally compact groups and our methods are adapted to chiral theories
on the circle.
