{"title":"Non-linear traces on the algebra of compact operators and majorization","authors":"Masaru Nagisa , Yasuo Watatani","doi":"10.1016/j.indag.2023.02.002","DOIUrl":null,"url":null,"abstract":"<div><p><span>We study non-linear traces of the Choquet type and the Sugeno type on the algebra of compact operators. They have certain partial additivities. We show that these partial additivities characterize non-linear traces of both the Choquet type and the Sugeno type respectively. There exists a close relation between non-linear traces of the Choquet type and majorization theory. We study trace class operators for non-linear traces of the Choquet type. More generally we discuss Schatten–von Neumann </span><span><math><mi>p</mi></math></span><span>-class operators for non-linear traces of the Choquet type. We determine when they form Banach spaces<span>. This is an attempt at non-commutative integration theory for non-linear traces of the Choquet type on the algebra of compact operators. We also consider the triangle inequality for non-linear traces of the Sugeno type.</span></span></p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":null,"pages":null},"PeriodicalIF":0.5000,"publicationDate":"2023-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Indagationes Mathematicae-New Series","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0019357723000198","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We study non-linear traces of the Choquet type and the Sugeno type on the algebra of compact operators. They have certain partial additivities. We show that these partial additivities characterize non-linear traces of both the Choquet type and the Sugeno type respectively. There exists a close relation between non-linear traces of the Choquet type and majorization theory. We study trace class operators for non-linear traces of the Choquet type. More generally we discuss Schatten–von Neumann -class operators for non-linear traces of the Choquet type. We determine when they form Banach spaces. This is an attempt at non-commutative integration theory for non-linear traces of the Choquet type on the algebra of compact operators. We also consider the triangle inequality for non-linear traces of the Sugeno type.
研究紧算子代数上Choquet型和Sugeno型的非线性迹。它们有部分可加性。我们证明了这些部分可加性分别表征了Choquet型和Sugeno型的非线性轨迹。Choquet型的非线性轨迹与多数化理论之间存在着密切的关系。研究了Choquet型非线性迹的迹类算子。更一般地,我们讨论了Choquet型的非线性轨迹的schaten - von Neumann p类算子。我们确定它们何时形成巴拿赫空间。这是对紧算子代数上的非线性Choquet型迹的非交换积分理论的一个尝试。我们还考虑了Sugeno型非线性轨迹的三角形不等式。
期刊介绍:
Indagationes Mathematicae is a peer-reviewed international journal for the Mathematical Sciences of the Royal Dutch Mathematical Society. The journal aims at the publication of original mathematical research papers of high quality and of interest to a large segment of the mathematics community. The journal also welcomes the submission of review papers of high quality.