{"title":"关于测度的对数","authors":"H. Raubenheimer, J. van Appel","doi":"10.1007/s11117-023-01015-2","DOIUrl":null,"url":null,"abstract":"Abstract Let A be a Banach algebra and let $$x\\in A$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>A</mml:mi> </mml:mrow> </mml:math> have the property that its spectrum does not separate 0 from infinity. It is well known that x has a logarithm, i.e., there exists $$y\\in A$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>y</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>A</mml:mi> </mml:mrow> </mml:math> with $$x=e^y$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>e</mml:mi> <mml:mi>y</mml:mi> </mml:msup> </mml:mrow> </mml:math> . We will use this statement to identify measures defined on a locally compact group to have logarithms. Also, we will show that the converse of the above statement is in general not true. Our results will be related to infinitely divisible probability measures.","PeriodicalId":54596,"journal":{"name":"Positivity","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2023-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On logarithms of measures\",\"authors\":\"H. Raubenheimer, J. van Appel\",\"doi\":\"10.1007/s11117-023-01015-2\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract Let A be a Banach algebra and let $$x\\\\in A$$ <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\"> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>A</mml:mi> </mml:mrow> </mml:math> have the property that its spectrum does not separate 0 from infinity. It is well known that x has a logarithm, i.e., there exists $$y\\\\in A$$ <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\"> <mml:mrow> <mml:mi>y</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>A</mml:mi> </mml:mrow> </mml:math> with $$x=e^y$$ <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\"> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>e</mml:mi> <mml:mi>y</mml:mi> </mml:msup> </mml:mrow> </mml:math> . We will use this statement to identify measures defined on a locally compact group to have logarithms. Also, we will show that the converse of the above statement is in general not true. Our results will be related to infinitely divisible probability measures.\",\"PeriodicalId\":54596,\"journal\":{\"name\":\"Positivity\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2023-10-16\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Positivity\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1007/s11117-023-01015-2\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Positivity","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s11117-023-01015-2","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
摘要设A是一个Banach代数，且设$$x\in A$$ x∈A具有其谱不将0与无穷分开的性质。众所周知，x有对数，即存在$$y\in A$$ y∈a，且$$x=e^y$$ x = ey。我们将使用这个语句来确定在局部紧群上定义的具有对数的测度。同样，我们将证明上述陈述的反面通常是不正确的。我们的结果将与无限可分的概率测度有关。
Abstract Let A be a Banach algebra and let $$x\in A$$ x∈A have the property that its spectrum does not separate 0 from infinity. It is well known that x has a logarithm, i.e., there exists $$y\in A$$ y∈A with $$x=e^y$$ x=ey . We will use this statement to identify measures defined on a locally compact group to have logarithms. Also, we will show that the converse of the above statement is in general not true. Our results will be related to infinitely divisible probability measures.
期刊介绍：
The purpose of Positivity is to provide an outlet for high quality original research in all areas of analysis and its applications to other disciplines having a clear and substantive link to the general theme of positivity. Specifically, articles that illustrate applications of positivity to other disciplines - including but not limited to - economics, engineering, life sciences, physics and statistical decision theory are welcome.
The scope of Positivity is to publish original papers in all areas of mathematics and its applications that are influenced by positivity concepts.
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