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Quasi-states and the Poisson bracket on surfaces. (English) Zbl 1131.53046
Summary: We present a convexity-type result concerning simple quasi-states on closed manifolds. As a corollary, an inequality emerges which relates the Poisson bracket to the measure of non-additivity of a simple quasi-state on a closed surface equipped with an area form. In addition, we prove that the uniform norm of the Poisson bracket of two functions on a surface is stable from below under \(C^0\)-perturbations.

53D20 Momentum maps; symplectic reduction
28C15 Set functions and measures on topological spaces (regularity of measures, etc.)
53D17 Poisson manifolds; Poisson groupoids and algebroids
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