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Any continuous piecewise affine map α : Y ⊂ H → G (where Y is an element of the open coset ring) induces a completely bounded homomorphism Φα : A(G) → B(H) by setting Φαu = u•α on Y and Φαu = 0 off of Y. We show that if G is amenable then any completely bounded homomorphism Φ : A(G) → B(H) is of this form; and this theorem fails if G contains a discrete nonabelian free group. Our result generalises results of P.J. Cohen [8], B. Host [20] and of the first author [23]. We also obtain a description of all the idempotents in the Fourier-Stieltjes algebras which are contractive or positive definite.","publication_date":"2005,,","publication_name":"Journal of Functional Analysis","grobid_abstract_attachment_id":"105894193"},"document_type":"paper","pre_hit_view_count_baseline":null,"quality":"high","language":"en","title":"Completely bounded homomorphisms of the Fourier algebras","broadcastable":false,"draft":null,"has_indexable_attachment":true,"indexable":true}}["work"]; window.loswp.workCoauthors = [239478325]; window.loswp.locale = "en"; window.loswp.countryCode = "SG"; window.loswp.cwvAbTestBucket = ""; window.loswp.designVariant = "ds_vanilla"; window.loswp.fullPageMobileSutdModalVariant = "full_page_mobile_sutd_modal"; 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 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class="ds-related-work--abstract ds2-5-body-sm">Let G be a compact group. 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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="4" 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. 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The module action of A(G) of ℒ∞(G) is defined by the following: for ƒ ∈ A(G), φ ℒ∞(G), ƒ. φ ∈ ℒ∞(G) by 〈g, ƒ . φ〉 = 〈 ƒg, φ〉, g ∈ A(G) Also ‖ƒ . φ‖∞ ≥ ‖ƒ‖A ‖φ‖∞.</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":"Subalgebras of the dual of the Fourier algebra of a compact group","attachmentId":72832378,"attachmentType":"pdf","work_url":"https://www.academia.edu/58404997/Subalgebras_of_the_dual_of_the_Fourier_algebra_of_a_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/58404997/Subalgebras_of_the_dual_of_the_Fourier_algebra_of_a_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="5" data-entity-id="91674681" 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/91674681/On_the_fourier_algebra_of_certain_hypergroups">On the fourier algebra of certain hypergroups</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="49798333" href="https://independent.academia.edu/SeyedMahmoudManjegani">Seyed Mahmoud Manjegani</a></div><p class="ds-related-work--metadata 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="6" data-entity-id="83847920" 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/83847920/Separating_maps_between_Lebesgue_Fourier_algebras">Separating maps between Lebesgue-Fourier 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="62104376" href="https://iut.academia.edu/RasoulNasrIsfahani">Rasoul Nasr-Isfahani</a></div><p class="ds-related-work--abstract ds2-5-body-sm">Let $G_{1}$ and $G_{2}$ be locally compact groups; it is known that ${\cal L}A(G_{i})=L^1(G_i)\cap A(G_i)$ is an abstract Segal algebra with respect to $A(G_{i})$ for $i = 1,2$. 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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="102998053" 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/102998053/On_ideals_in_the_bidual_of_the_Fourier_algebra_and_related_algebras">On ideals in the bidual of the Fourier algebra and related 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="272814050" href="https://independent.academia.edu/FilaliManal">Manal Filali</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Functional Analysis, 2010</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 ideals in the bidual of the Fourier algebra and related algebras","attachmentId":103121683,"attachmentType":"pdf","work_url":"https://www.academia.edu/102998053/On_ideals_in_the_bidual_of_the_Fourier_algebra_and_related_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/102998053/On_ideals_in_the_bidual_of_the_Fourier_algebra_and_related_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="73982366" 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/73982366/New_deformations_of_Convolution_algebras_and_Fourier_algebras_on_locally_compact_groups">New deformations of Convolution algebras and Fourier algebras 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="13534588" href="https://independent.academia.edu/HunHeeLee">Hun Hee Lee</a></div><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":"New deformations of Convolution algebras and Fourier algebras on locally compact groups","attachmentId":83819308,"attachmentType":"pdf","work_url":"https://www.academia.edu/73982366/New_deformations_of_Convolution_algebras_and_Fourier_algebras_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/73982366/New_deformations_of_Convolution_algebras_and_Fourier_algebras_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 class="ds-related-work--container js-wsj-grid-card" data-collection-position="9" 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></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":105894193,"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":105894193,"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_105894193" 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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href="https://www.academia.edu/58405002/Operators_on_the_Fourier_algebra_with_weakly_compact_extensions">Operators on the Fourier algebra with weakly compact extensions</a><div class="ds-related-work--metadata"><a class="js-related-work-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">Canadian Journal of Mathematics, 1974</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":"Operators on the Fourier algebra with weakly compact extensions","attachmentId":72832375,"attachmentType":"pdf","work_url":"https://www.academia.edu/58405002/Operators_on_the_Fourier_algebra_with_weakly_compact_extensions","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" 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data-author-id="42725303" href="https://colorado.academia.edu/ArlanRamsay">Arlan Ramsay</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Functional Analysis, 1997</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 Algebras of Locally Compact Groupoids","attachmentId":42154237,"attachmentType":"pdf","work_url":"https://www.academia.edu/21593854/Fourier_Stieltjes_Algebras_of_Locally_Compact_Groupoids","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-related-work-grid-card-view-pdf" href="https://www.academia.edu/21593854/Fourier_Stieltjes_Algebras_of_Locally_Compact_Groupoids"><span 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Forrest</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Functional Analysis, 2003</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":"Ideals with bounded approximate identities in Fourier algebras","attachmentId":39572923,"attachmentType":"pdf","work_url":"https://www.academia.edu/17547420/Ideals_with_bounded_approximate_identities_in_Fourier_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-related-work-grid-card-view-pdf" href="https://www.academia.edu/17547420/Ideals_with_bounded_approximate_identities_in_Fourier_algebras"><span class="ds2-5-text-link__content">View PDF</span><span 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href="https://www.academia.edu/73982367/Similarity_degree_of_Fourier_algebras">Similarity degree of Fourier algebras</a><div class="ds-related-work--metadata"><a class="js-related-work-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">Journal of Functional Analysis, 2016</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":"Similarity degree of Fourier algebras","attachmentId":82302159,"attachmentType":"pdf","work_url":"https://www.academia.edu/73982367/Similarity_degree_of_Fourier_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 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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-related-work-grid-card-view-pdf" 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class="ds-related-work--metadata ds2-5-body-xs">Pacific Journal of Mathematics, 1990</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":"Amenability of discrete convolution algebras, the commutative case","attachmentId":94968522,"attachmentType":"pdf","work_url":"https://www.academia.edu/91778090/Amenability_of_discrete_convolution_algebras_the_commutative_case","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-related-work-grid-card-view-pdf" href="https://www.academia.edu/91778090/Amenability_of_discrete_convolution_algebras_the_commutative_case"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" 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class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="20" data-entity-id="7964866" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/7964866/SOME_EXTREMELY_AMENABLE_GROUPS_RELATED_TO_OPERATOR_ALGEBRAS_AND_ERGODIC_THEORY">SOME EXTREMELY AMENABLE GROUPS RELATED TO OPERATOR ALGEBRAS AND ERGODIC THEORY</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="15186350" href="https://independent.academia.edu/TGiordano">Thierry Giordano</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of The Institute of Mathematics of Jussieu, 2007</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 EXTREMELY AMENABLE GROUPS RELATED 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