REZA ESMAILVANDI, MEHDI NEMATI, NAGESWARAN SHRAVAN KUMAR
{"title":"ON THE ALGEBRAS","authors":"REZA ESMAILVANDI, MEHDI NEMATI, NAGESWARAN SHRAVAN KUMAR","doi":"10.1017/s1446788722000192","DOIUrl":null,"url":null,"abstract":"<p>Let <span>H</span> be an ultraspherical hypergroup and let <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231109001332944-0291:S1446788722000192:S1446788722000192_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$A(H)$</span></span></img></span></span> be the Fourier algebra associated with <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231109001332944-0291:S1446788722000192:S1446788722000192_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$H.$</span></span></img></span></span> In this paper, we study the dual and the double dual of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231109001332944-0291:S1446788722000192:S1446788722000192_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$A(H).$</span></span></img></span></span> We prove among other things that the subspace of all uniformly continuous functionals on <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231109001332944-0291:S1446788722000192:S1446788722000192_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$A(H)$</span></span></img></span></span> forms a <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231109001332944-0291:S1446788722000192:S1446788722000192_inline8.png\"><span data-mathjax-type=\"texmath\"><span>$C^*$</span></span></img></span></span>-algebra. We also prove that the double dual <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231109001332944-0291:S1446788722000192:S1446788722000192_inline9.png\"><span data-mathjax-type=\"texmath\"><span>$A(H)^{\\ast \\ast }$</span></span></img></span></span> is neither commutative nor semisimple with respect to the Arens product, unless the underlying hypergroup <span>H</span> is finite. Finally, we study the unit elements of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231109001332944-0291:S1446788722000192:S1446788722000192_inline10.png\"><span data-mathjax-type=\"texmath\"><span>$A(H)^{\\ast \\ast }.$</span></span></img></span></span></p>","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.5000,"publicationDate":"2022-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of the Australian Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/s1446788722000192","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Let H be an ultraspherical hypergroup and let $A(H)$ be the Fourier algebra associated with $H.$ In this paper, we study the dual and the double dual of $A(H).$ We prove among other things that the subspace of all uniformly continuous functionals on $A(H)$ forms a $C^*$-algebra. We also prove that the double dual $A(H)^{\ast \ast }$ is neither commutative nor semisimple with respect to the Arens product, unless the underlying hypergroup H is finite. Finally, we study the unit elements of $A(H)^{\ast \ast }.$
期刊介绍：
The Journal of the Australian Mathematical Society is the oldest journal of the Society, and is well established in its coverage of all areas of pure mathematics and mathematical statistics. It seeks to publish original high-quality articles of moderate length that will attract wide interest. Papers are carefully reviewed, and those with good introductions explaining the meaning and value of the results are preferred.
Published Bi-monthly
Published for the Australian Mathematical Society
$A(H)$ be the Fourier algebra associated with 
$H.$ In this paper, we study the dual and the double dual of 
$A(H).$ We prove among other things that the subspace of all uniformly continuous functionals on 
$A(H)$ forms a 
$C^*$-algebra. We also prove that the double dual 
$A(H)^{\ast \ast }$ is neither commutative nor semisimple with respect to the Arens product, unless the underlying hypergroup H is finite. Finally, we study the unit elements of 
$A(H)^{\ast \ast }.$
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