On almost Poisson commutativity in dimension two
The author proves that \(n\) smooth functions on a closed \(n\)-dimensional manifold with a volume form, whose bracket is small (in the \(L^1\)-norm), can be approximated in the uniform norm by commuting smooth functions. For the particular case of a closed surface with an area form, two continuous functions Poisson commute if and only if they can be approximated in the uniform norm by Poisson commuting smooth functions. Moreover, a consequence of this in the theory of quasi-states is that, in this case, a quasi-state is linear on Poisson commutative subspaces of all smooth functions.
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