  The theory of alpha_star-cohomology is studied thoroughly and it is shown
that in each cohomology class there exists a unique 2-cocycle, the harmonic
form, which generates a particular Groenewold-Moyal star product. This leads to
an algebraic classification of translation-invariant non-commutative structures
and shows that any general translationinvariant non-commutative quantum field
theory is physically equivalent to a Groenewold- Moyal non-commutative quantum
field theory.
