{"title":"离散群的双面庞培问题","authors":"P. Linnell, M. Puls","doi":"10.1090/bproc/124","DOIUrl":null,"url":null,"abstract":"<p>We consider a two-sided Pompeiu type problem for a discrete group <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\n <mml:semantics>\n <mml:mi>G</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. We give necessary and sufficient conditions for a finite subset <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\">\n <mml:semantics>\n <mml:mi>K</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">K</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\n <mml:semantics>\n <mml:mi>G</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> to have the <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper F left-parenthesis upper G right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">F</mml:mi>\n </mml:mrow>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>G</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mathcal {F}(G)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-Pompeiu property. Using group von Neumann algebra techniques, we give necessary and sufficient conditions for <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\n <mml:semantics>\n <mml:mi>G</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> to be an <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script l squared left-parenthesis upper G right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:msup>\n <mml:mi>ℓ<!-- ℓ --></mml:mi>\n <mml:mn>2</mml:mn>\n </mml:msup>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>G</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\ell ^2(G)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-Pompeiu group.</p>","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"11 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2020-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The two-sided Pompeiu problem for discrete groups\",\"authors\":\"P. Linnell, M. Puls\",\"doi\":\"10.1090/bproc/124\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We consider a two-sided Pompeiu type problem for a discrete group <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper G\\\">\\n <mml:semantics>\\n <mml:mi>G</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">G</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>. We give necessary and sufficient conditions for a finite subset <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper K\\\">\\n <mml:semantics>\\n <mml:mi>K</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">K</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> of <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper G\\\">\\n <mml:semantics>\\n <mml:mi>G</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">G</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> to have the <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"script upper F left-parenthesis upper G right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi class=\\\"MJX-tex-caligraphic\\\" mathvariant=\\\"script\\\">F</mml:mi>\\n </mml:mrow>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>G</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathcal {F}(G)</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>-Pompeiu property. Using group von Neumann algebra techniques, we give necessary and sufficient conditions for <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper G\\\">\\n <mml:semantics>\\n <mml:mi>G</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">G</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> to be an <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"script l squared left-parenthesis upper G right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:msup>\\n <mml:mi>ℓ<!-- ℓ --></mml:mi>\\n <mml:mn>2</mml:mn>\\n </mml:msup>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>G</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\ell ^2(G)</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>-Pompeiu group.</p>\",\"PeriodicalId\":106316,\"journal\":{\"name\":\"Proceedings of the American Mathematical Society, Series B\",\"volume\":\"11 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2020-09-28\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Proceedings of the American Mathematical Society, Series B\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1090/bproc/124\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the American Mathematical Society, Series B","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/bproc/124","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
We consider a two-sided Pompeiu type problem for a discrete group GG. We give necessary and sufficient conditions for a finite subset KK of GG to have the F(G)\mathcal {F}(G)-Pompeiu property. Using group von Neumann algebra techniques, we give necessary and sufficient conditions for GG to be an ℓ2(G)\ell ^2(G)-Pompeiu group.
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