{"title":"Tracial weights on topological graph algebras","authors":"JOHANNES CHRISTENSEN","doi":"10.1017/etds.2024.20","DOIUrl":null,"url":null,"abstract":"<p>We describe two kinds of regular invariant measures on the boundary path space <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240304092108222-0628:S0143385724000208:S0143385724000208_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$\\partial E$</span></span></img></span></span> of a second countable topological graph <span>E</span>, which allows us to describe all extremal tracial weights on <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240304092108222-0628:S0143385724000208:S0143385724000208_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$C^{*}(E)$</span></span></img></span></span> which are not gauge-invariant. Using this description, we prove that all tracial weights on the C<span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240304092108222-0628:S0143385724000208:S0143385724000208_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$^{*}$</span></span></img></span></span>-algebra <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240304092108222-0628:S0143385724000208:S0143385724000208_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$C^{*}(E)$</span></span></img></span></span> of a second countable topological graph <span>E</span> are gauge-invariant when <span>E</span> is free. This in particular implies that all tracial weights on <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240304092108222-0628:S0143385724000208:S0143385724000208_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$C^{*}(E)$</span></span></img></span></span> are gauge-invariant when <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240304092108222-0628:S0143385724000208:S0143385724000208_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$C^{*}(E)$</span></span></img></span></span> is simple and separable.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/etds.2024.20","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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Abstract
We describe two kinds of regular invariant measures on the boundary path space $\partial E$ of a second countable topological graph E, which allows us to describe all extremal tracial weights on $C^{*}(E)$ which are not gauge-invariant. Using this description, we prove that all tracial weights on the C$^{*}$-algebra $C^{*}(E)$ of a second countable topological graph E are gauge-invariant when E is free. This in particular implies that all tracial weights on $C^{*}(E)$ are gauge-invariant when $C^{*}(E)$ is simple and separable.
$\partial E$ of a second countable topological graph E, which allows us to describe all extremal tracial weights on 
$C^{*}(E)$ which are not gauge-invariant. Using this description, we prove that all tracial weights on the C
$^{*}$-algebra 
$C^{*}(E)$ of a second countable topological graph E are gauge-invariant when E is free. This in particular implies that all tracial weights on 
$C^{*}(E)$ are gauge-invariant when 
$C^{*}(E)$ is simple and separable.
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