Amplified graph C*-algebras II: Reconstruction

Proceedings of the American Mathematical Society, Series B Pub Date : 2020-07-02 DOI:10.1090/bproc/112

S. Eilers, Efren Ruiz, A. Sims

{"title":"Amplified graph C*-algebras II: Reconstruction","authors":"S. Eilers, Efren Ruiz, A. Sims","doi":"10.1090/bproc/112","DOIUrl":null,"url":null,"abstract":"<p>Let <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper E\">\n <mml:semantics>\n <mml:mi>E</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">E</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> be a countable directed graph that is amplified in the sense that whenever there is an edge from <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"v\">\n <mml:semantics>\n <mml:mi>v</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">v</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> to <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"w\">\n <mml:semantics>\n <mml:mi>w</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">w</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, there are infinitely many edges from <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"v\">\n <mml:semantics>\n <mml:mi>v</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">v</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> to <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"w\">\n <mml:semantics>\n <mml:mi>w</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">w</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. We show that <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper E\">\n <mml:semantics>\n <mml:mi>E</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">E</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> can be recovered from <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C Superscript asterisk Baseline left-parenthesis upper E right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:msup>\n <mml:mi>C</mml:mi>\n <mml:mo>∗</mml:mo>\n </mml:msup>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>E</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">C^*(E)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> together with its canonical gauge-action, and also from <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L Subscript double-struck upper K Baseline left-parenthesis upper E right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mi>L</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">K</mml:mi>\n </mml:mrow>\n </mml:msub>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>E</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">L_\\mathbb {K}(E)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> together with its canonical grading.</p>","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"57 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2020-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"6","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the American Mathematical Society, Series B","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/bproc/112","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}

引用次数: 6

Abstract

Let E E be a countable directed graph that is amplified in the sense that whenever there is an edge from v v to w w , there are infinitely many edges from v v to w w . We show that E E can be recovered from C ∗ ( E ) C^*(E) together with its canonical gauge-action, and also from L K ( E ) L_\mathbb {K}(E) together with its canonical grading.