Quasi-states and quasi-measures

Quasi-states and symplectic intersections

“…The relation between a quasi-state ζ and the corresponding quasi-measure τ ζ is the following [3]. Given a closed X ⊂ M, consider the set F X of smooth functions M → [0, 1] which are identically equal to 1 on X.…”

“…According to the general construction from [3], any quasi-measure τ gives rise to a quasi-state ζ τ . Roughly speaking, the definition of ζ τ is as follows.…”

“…In this paper we show that such rigidity phenomena in a closed symplectic manifold M sometimes formally follow from the existence of real-valued functionals with some interesting algebraic properties on the Poisson algebra C ∞ (M). On the one hand, these functionals are related to the notions of quasi-state and quasi-measure (which have been recently called topological measures) on M (see Section 3) which originate in quantum mechanics [1], [2] and have been a subject of intensive study in recent years following the paper [3] by J.F.Aarnes.…”

Py-calabi quasi-morphisms and quasi-states on orientable surfaces of higher genus

“…The notion of a quasi-state originates in quantom mechanics [1,2], and has been a subject of intensive study in recent years following the paper [3] by J. F. Aarnes. Here is the definition.…”

Simple topological measures and a lifting problem

“…These are exactly the topological measures that are subadditive. The important discovery of Aarnes was the existence of a non-Borel topological measure [1]. With the help of an integration theory for continuous functions, this non-Borel topological measure gave an example of a non-linear quasi-state on a commutative C * -algebra.…”