The notion of a quasi-state, whose origins lie in quantum mechanics, was introduced by an analyst Johan Aarnes. It can be roughly described as an "almost linear" positive functional on C(X), where X is a compact metric space. A quasi-measure is an "almost measure" on X corresponding to a quasi-state by Aarnes' extension of the classical Riesz representation theorem.
I will discuss how quasi-states and quasi-measures appear in symplectic topology as a convenient package for a certain information contained in the Hamiltonian Floer theory and how these and similar objects can be used to prove results on symplectic rigidity, including non-displaceability by a Hamiltonian isotopy of some Lagrangian fibers in symplectic toric manifolds.
The talk is based on joint works with P.Biran, L.Polterovich and F.Zapolsky.