Approximation of quasi-states on manifolds

Abstract

Quasi-states are certain not necessarily linear functionals on the space of continuous functions on a compact Hausdorff space. They were discovered as a part of an attempt to understand the axioms of quantum mechanics due to von Neumann. A very interesting and fundamental example is given by the so-called median quasi-state on S 2 . In this paper we present an algorithm which numerically computes it to any specified accuracy. The error estimate of the algorithm crucially relies on metric continuity properties of a map, which constructs quasi-states from probability measures, with respect to appropriate Wasserstein metrics. We close with non-approximation results, particularly for symplectic quasi-states.

Weak Convergence of Topological Measures

“…If a deficient topological measure corresponds to R, then μ ≤ M. Thus, the family M is uniformly bounded in variation, and we may use the same argument as in the previous part. 6) is compact by the Arzela-Ascoli theorem, one can basically repeat an argument from [21,Proposition 1.10] to show that the weak convergence of (μ α ) to μ implies convergence in the metric d KR . A finite Radon measure on a compact space is a regular Borel measure, so our definition of a proper deficient topological measure (which is given in [14]) coincides with definitions in papers prior to [14].…”

Quasi-linear functionals on locally compact spaces

“…, g with support contained in K. Theorem 4.5 and Corollary 4.10 say that ρ is Lipschitz continuous. Lipschitz continuity is used in many results in symplectic geometry; it is also used in the proof of results involving the Kantorovich-Rubinstein metric for (deficient) topological measures (see [22], [18]).…”

“…In [22] the authors present a wide class of quasi-states that can not be approximated by specific quasi-states. Symplectic quasi-states constructed via Floer homology on symplectic manifolds of dimension at least 4 can not be approximated by quasi-states corresponding to simple topological measures obtained as compositions of measures with a q-function that takes only two values.…”

“…(A quasi-linear functional corresponding to a simple topological measure is multiplicative on singly generated subalgebras, see [4], [20]). From the point of view of the theory of topological measures, the explanation of the "non-approximation" results from [22] is suggested by [10,Cor. 4.13] or by the fact that approximation of a quasi-state, in general, requires a much larger collection than just quasi-integrals corresponding to simple topological measures.…”

A characterization of heaviness in terms of relative symplectic cohomology

“…More recently, Dickstein–Ganor–Polterovich–Zapolsky [4] introduced their quantum cohomology ideal‐valued quasi‐measures. For example, a compact set has full measure if and only if it is SH‐full (see [4, Remark 1.46]). They also defined a new notion called SH‐heavy and conjectured.…”