Quasi-States, Quasi-Morphisms, and the Moment Map

We prove that symplectic quasi-states and quasi-morphisms on a symplectic manifold descend under symplectic reduction on a superheavy level set of a Hamiltonian torus action. Using a construction due to Abreu and Macarini, in each dimension, at least four, we produce a closed symplectic toric manifold with infinite-dimensional spaces of symplectic quasi-states and quasi-morphisms, and a one-parameter family of nondisplaceable Lagrangian tori. By using McDuff's method of probes, we also show how Ostrover and Tyomkin's method for finding distinct spectral quasi-states in symplectic toric Fano manifolds can also be used to find different superheavy toric fibers.