The two-sided Pompeiu problem for discrete groups

Proceedings of the American Mathematical Society, Series B Pub Date : 2020-09-28 DOI:10.1090/bproc/124

P. Linnell, M. Puls

{"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</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}

引用次数: 0

Abstract

We consider a two-sided Pompeiu type problem for a discrete group G G . We give necessary and sufficient conditions for a finite subset K K of G G to have the F ( G ) \mathcal {F}(G) -Pompeiu property. Using group von Neumann algebra techniques, we give necessary and sufficient conditions for G G to be an ℓ 2 ( G ) \ell ^2(G) -Pompeiu group.