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Though, in contrast to standard quantum field theory (QFT), the tensor-algebra over the test-function space of hyperfunction quantum field theory has no local structure, the localization properties of states on this algebra can be used to derive asymptotic Abelianness in spacelike directions. Again, in contrast to standard QFT, the closure of (Hermitian) field operators can destroy localization properties. This problem is addressed in a natural modification of the definition of the closure, called the local closure. This allows one, in conjunction with asymptotic Abelianness, to define a proper reduction of the field algebra to the subspace of the translation invariant states, and to investigate the dimension of this subspace. © 1998 American Institute of Physics.
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