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compact group that has an infinite abelian subgroup. This extends the theorems of Helson, Kahane, Katznelson, and Rudin [4] which apply to the algebra of absolutely convergent Fourier series on compact abelian groups. The Fourier algebra of a locally compact group has been studied by H. Mirkil [ó], W. F. Stinespring [9], R. A. Mayer [5], C. Herz [3], and most thoroughly by P. Eymard [l]. We will state here the relevant definitions and facts, and prove that the restriction of the Fourier algebra to a closed subgroup is the Fourier algebra of the subgroup, and use this to lift up the theorem on operating functions.","publication_date":"1969,,","publication_name":"Proceedings of the American Mathematical Society","grobid_abstract_attachment_id":"72838146"},"document_type":"paper","pre_hit_view_count_baseline":null,"quality":"high","language":"en","title":"Functions that operate in the Fourier algebra of a compact group","broadcastable":true,"draft":null,"has_indexable_attachment":true,"indexable":true}}["work"]; window.loswp.workCoauthors = [39545224]; window.loswp.locale = "en"; window.loswp.countryCode = "SG"; window.loswp.cwvAbTestBucket = ""; window.loswp.designVariant = "ds_vanilla"; window.loswp.fullPageMobileSutdModalVariant = "control"; window.loswp.useOptimizedScribd4genScript = false; window.loswp.appleClientId = 'edu.academia.applesignon';</script><script defer="" src="https://accounts.google.com/gsi/client"></script><div class="ds-loswp-container"><div class="ds-work-card--grid-container"><div class="ds-work-card--container js-loswp-work-card"><div class="ds-work-card--cover"><div class="ds-work-cover--wrapper"><div class="ds-work-cover--container"><button class="ds-work-cover--clickable js-swp-download-button" data-signup-modal="{"location":"swp-splash-paper-cover","attachmentId":72838146,"attachmentType":"pdf"}"><img alt="First page of “Functions that operate in the Fourier algebra of a compact group”" class="ds-work-cover--cover-thumbnail" src="https://0.academia-photos.com/attachment_thumbnails/72838146/mini_magick20211016-14879-121379s.png?1634401744" /><img alt="PDF Icon" class="ds-work-cover--file-icon" src="//a.academia-assets.com/assets/single_work_splash/adobe.icon-574afd46eb6b03a77a153a647fb47e30546f9215c0ee6a25df597a779717f9ef.svg" /><div class="ds-work-cover--hover-container"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span><p>Download Free PDF</p></div><div class="ds-work-cover--ribbon-container">Download Free PDF</div><div class="ds-work-cover--ribbon-triangle"></div></button></div></div></div><div class="ds-work-card--work-information"><h1 class="ds-work-card--work-title">Functions that operate in the Fourier algebra of a compact group</h1><div class="ds-work-card--work-authors ds-work-card--detail"><a class="ds-work-card--author js-wsj-grid-card-author ds2-5-body-md ds2-5-body-link" data-author-id="39545224" href="https://independent.academia.edu/CharlesDunkl"><img alt="Profile image of Charles Dunkl" class="ds-work-card--author-avatar" src="//a.academia-assets.com/images/s65_no_pic.png" />Charles Dunkl</a></div><div class="ds-work-card--detail"><p class="ds-work-card--detail ds2-5-body-sm">1969, Proceedings of the American Mathematical Society</p><div class="ds-work-card--work-metadata"><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">visibility</span><p class="ds2-5-body-sm" id="work-metadata-view-count">…</p></div><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">description</span><p class="ds2-5-body-sm">5 pages</p></div><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">link</span><p class="ds2-5-body-sm">1 file</p></div></div><script>(async () => { const workId = 58405037; 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For 1≤ p≤∞ we introduce a class of Banach function algebras A^p(G) on G which are the Fourier algebras in the case p=1, and for p=2 are certain algebras discovered in forrestss1. In the case p=2 we find that A^p(G)A^p(H) if and only if G and H are isomorphic compact groups. These algebras admit natural operator space structures, and also weighted versions, which we call p-Beurling-Fourier algebras. We study various amenability and operator amenability properties, Arens regularity and representability as operator algebras. For a connected Lie G and p&gt;1, our techniques of estimation of when certain p-Beurling-Fourier algebras are operator algebras rely more on the fine structure of G, than in the case p=1. We also study restrictions to subgroups. In the case that G=SU(2), restrict to a torus and obtain some exotic algebras of Laurent series. We study amenability properties of these new algebras, as well.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"p-Fourier algebras on compact groups","attachmentId":82302099,"attachmentType":"pdf","work_url":"https://www.academia.edu/73982382/p_Fourier_algebras_on_compact_groups","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/73982382/p_Fourier_algebras_on_compact_groups"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="1" data-entity-id="16400216" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/16400216/On_Twisted_Fourier_Analysis_and_Convergence_of_Fourier_Series_on_Discrete_Groups">On Twisted Fourier Analysis and Convergence of Fourier Series on Discrete Groups</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="35538661" href="https://uniroma1.academia.edu/RobertoConti">Roberto Conti</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Fourier Analysis and Applications, 2009</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"On Twisted Fourier Analysis and Convergence of Fourier Series on Discrete Groups","attachmentId":42500779,"attachmentType":"pdf","work_url":"https://www.academia.edu/16400216/On_Twisted_Fourier_Analysis_and_Convergence_of_Fourier_Series_on_Discrete_Groups","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/16400216/On_Twisted_Fourier_Analysis_and_Convergence_of_Fourier_Series_on_Discrete_Groups"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="2" data-entity-id="58404997" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/58404997/Subalgebras_of_the_dual_of_the_Fourier_algebra_of_a_compact_group">Subalgebras of the dual of the Fourier algebra of a compact group</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="39545224" href="https://independent.academia.edu/CharlesDunkl">Charles Dunkl</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Mathematical Proceedings of the Cambridge Philosophical Society, 1972</p><p class="ds-related-work--abstract ds2-5-body-sm">We let G denote an infinite compact group and G its dual. We use the notation of our book ((l), Chapters 7 and 8). Recall A(G) denotes the Fourier algebra of G (an algebra of continuous functions on G), and ℒ∞(G) denotes its dual space under the pairing 〈ƒ,φ〉 (ƒ ∈ A(G), φ ∈ ℒ∞(G)). Further, note ℒ∞(G) is identified with the C*-algebra of bounded operators on L2(G) commuting with left translation. 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ds2-5-body-xs">Quaestiones Mathematicae, 2019</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"On the fourier algebra of certain hypergroups","attachmentId":94895220,"attachmentType":"pdf","work_url":"https://www.academia.edu/91674681/On_the_fourier_algebra_of_certain_hypergroups","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/91674681/On_the_fourier_algebra_of_certain_hypergroups"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="7" data-entity-id="116584436" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/116584436/Some_Beurling_Fourier_algebras_on_compact_groups_are_operator_algebras">Some Beurling–Fourier algebras on compact groups are operator algebras</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="13534588" href="https://independent.academia.edu/HunHeeLee">Hun Hee Lee</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Transactions of the American Mathematical Society, 2015</p><p class="ds-related-work--abstract ds2-5-body-sm">Let G G be a compact connected Lie group. The question of when a weighted Fourier algebra on G G is completely isomorphic to an operator algebra will be investigated in this paper. We will demonstrate that the dimension of the group plays an important role in the question. More precisely, we will get a positive answer to the question when we consider a polynomial type weight coming from a length function on G G with the order of growth strictly bigger than half of the dimension of the group. The case of S U ( n ) SU(n) will be examined, focusing more on the details including negative results. The proof for the positive directions depends on a non-commutative version of the Littlewood multiplier theory, which we will develop in this paper, and the negative directions will be taken care of by restricting to a maximal torus.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Some Beurling–Fourier algebras on compact groups are operator algebras","attachmentId":112673808,"attachmentType":"pdf","work_url":"https://www.academia.edu/116584436/Some_Beurling_Fourier_algebras_on_compact_groups_are_operator_algebras","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/116584436/Some_Beurling_Fourier_algebras_on_compact_groups_are_operator_algebras"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="8" data-entity-id="109451115" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/109451115/Fundamental_Theorems_of_Fourier_Stieltjes_Transform_Defined_by_Induced_Representation_on_Locally_Compact_Group">Fundamental Theorems of Fourier-Stieltjes Transform Defined by Induced Representation on Locally Compact Group</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="137244740" href="https://independent.academia.edu/Hounkonnou">Norbert Hounkonnou</a></div><p class="ds-related-work--metadata ds2-5-body-xs">arXiv (Cornell University), 2022</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Fundamental Theorems of Fourier-Stieltjes Transform Defined by Induced Representation on Locally Compact Group","attachmentId":107572730,"attachmentType":"pdf","work_url":"https://www.academia.edu/109451115/Fundamental_Theorems_of_Fourier_Stieltjes_Transform_Defined_by_Induced_Representation_on_Locally_Compact_Group","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/109451115/Fundamental_Theorems_of_Fourier_Stieltjes_Transform_Defined_by_Induced_Representation_on_Locally_Compact_Group"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="9" data-entity-id="86094949" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/86094949/Fourier_Stieltjes_transform_defined_by_induced_representation_on_locally_compact_groups">Fourier-Stieltjes transform defined by induced representation on locally compact groups</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="137244740" href="https://independent.academia.edu/Hounkonnou">Norbert Hounkonnou</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2022</p><p class="ds-related-work--abstract ds2-5-body-sm">Abstract. In this work we extend the Fourier-Stieltjes transform of a vector measure and a continuous function defined on compact groups to locally compact groups. To do so, we consider a representation L of a normal compact subgroup K of a locally compact group G, and we use a representation of G induced by that of L. Then, we define the Fourier-Stieltjes transform of a vector measure and that of a continuous function with compact support defined on G from the representation of G. Then, we extend the Shur orthogonality relation established for compact groups to locally compact groups by using the representations of G induced by the unitary representations of one of its normal compact subgroups. This extension enables us to develop a Fourier-Stieltjes transform in series form that is linear, continuous, and invertible.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Fourier-Stieltjes transform defined by induced representation on locally compact groups","attachmentId":90624887,"attachmentType":"pdf","work_url":"https://www.academia.edu/86094949/Fourier_Stieltjes_transform_defined_by_induced_representation_on_locally_compact_groups","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/86094949/Fourier_Stieltjes_transform_defined_by_induced_representation_on_locally_compact_groups"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div></div></div><div class="ds-sticky-ctas--wrapper js-loswp-sticky-ctas hidden"><div class="ds-sticky-ctas--grid-container"><div class="ds-sticky-ctas--container"><button class="ds2-5-button js-swp-download-button" data-signup-modal="{"location":"continue-reading-button--sticky-ctas","attachmentId":72838146,"attachmentType":"pdf","workUrl":null}">See full PDF</button><button class="ds2-5-button ds2-5-button--secondary js-swp-download-button" data-signup-modal="{"location":"download-pdf-button--sticky-ctas","attachmentId":72838146,"attachmentType":"pdf","workUrl":null}"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span>Download PDF</button></div></div></div><div class="ds-below-fold--grid-container"><div class="ds-work--container js-loswp-embedded-document"><div class="attachment_preview" data-attachment="Attachment_72838146" style="display: none"><div class="js-scribd-document-container"><div class="scribd--document-loading js-scribd-document-loader" style="display: block;"><img alt="Loading..." src="//a.academia-assets.com/images/loaders/paper-load.gif" /><p>Loading Preview</p></div></div><div style="text-align: center;"><div class="scribd--no-preview-alert js-preview-unavailable"><p>Sorry, preview is currently unavailable. 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