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Quasi-States and Symplectic Topology

Quasi-states are certain (not necessarily linear) functionals on the space of continuous functions of a compact space. We establish a connection between quasi-states and the Poisson bracket on certain closed symplectic manifolds, namely, we obtain an inequality relating the measure of non-additivity of a quasi-state on a pair of smooth functions and the C0-norm of their Poisson bracket. Applications are given to symplectic rigidity in the form of restrictions on partitions of unity, and to classical mechanics in the form of a lower bound on the error of simultaneous measurement of two observables. We prove a convexity result for simple quasi-states on arbitrary closed manifolds. As a corollary, the aforementioned inequality is generalized in the case of symplectic surfaces. Locality of the Poisson bracket on symplectic surfaces is proved. We construct examples of isotopy-invariant quasistates on surfaces of genus at least two. Lastly, we compute the value of Grubb’s quasi-state on the torus T2, which is the unique quasistate invariant under symplectomorphisms and vanishing on functions supported in a disk, and show that the restriction of this quasi-state to the set of smooth functions on the torus coincides with the functional stemming from a quasi-morphism on the identity component of the symplectomorphism group of the torus, due to Py.