On almost Poisson commutativity in dimension two
type: Item
instance of: scholarly article
title: On almost Poisson commutativity in dimension two
zbMATH Open document ID: 1208.53087
author: Frol Zapolsky
published in: Electronic Research Announcements in Mathematical Sciences
publication date: 2011-01-21T00:00:00Z
full work available at URL: https://arxiv.org/abs/1004.0870
review text: The author proves that \(n\) smooth functions on a closed \(n\)-dimensional manifold with a volume form, whose bracket is small (in the \(L^1\)-norm), can be approximated in the uniform norm by commuting smooth functions. For the particular case of a closed surface with an area form, two continuous functions Poisson commute if and only if they can be approximated in the uniform norm by Poisson commuting smooth functions. Moreover, a consequence of this in the theory of quasi-states is that, in this case, a quasi-state is linear on Poisson commutative subspaces of all smooth functions.
reviewed by: Adara M. Blaga
Mathematics Subject Classification ID: 53D17
Mathematics Subject Classification ID: 57M50
zbMATH DE Number: 5840314
zbMATH Keywords: Poisson bracket
zbMATH Keywords: surfaces
MaRDI profile type: Publication
OpenAlex ID: W2964241032
arXiv ID: 1004.0870
DOI: 10.3934/ERA.2010.17.155
description: scientific article