The theory of α*-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 translation-invariant non-commutative quantum field theory is physically equivalent to a Groenewold-Moyal non-commutative quantum field theory.

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The star product ⋆ is said to be complex if (fg)* = g*⋆f* for any
$f,g\in C^\infty (\mathbb {R}^m )$
f,gC(Rm)
.
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