On the uniqueness of the Moyal structure of phase-space functions

It is shown that the only associative algebras with a trivial center defined on functions of $\Rl^N$ by an integral kernel are generalized Moyal algebras, corresponding to some particular operator ordering. Similarly, the only such Lie algebras are generalized Moyal or Poisson Lie algebras. In both cases these structures are isomorphic respectively to the Moyal algebra, the Moyal Lie algebra and the Poisson Lie algebra. These results are independent of $N$, which is necessarily even, and generalize previous work on the subject.