On representations and topological aspects of positive maps on non-unital quasi *- algebras

On representations and topological aspects of positive maps on non-unital quasi *- algebras | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article On representations and topological aspects of positive maps on non-unital quasi *- algebras Giorgia Bellomonte, Bogdan Djordjevic, Stefan Ivkovich This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4402147/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 06 Sep, 2024 Read the published version in Positivity → Version 1 posted 9 You are reading this latest preprint version Abstract In this paper we provide a representation of a certain class of C*-valued positive sesquilinear and linear maps on non-unital quasi *- algebras, thus extending the results from [\cite{GBSICT}] to the case of non-unital quasi *-algebras. Also, we illustrate our results on the concrete examples of non-unital Banach quasi *-algebras, such as the standard Hilbert module over a commutative C*-algebra, Schatten p-ideals and noncommutative $L^2$-spaces induced by a semifinite, nonfinite trace. As a consequence of our results, we obtain a representation of all bounded positive linear C*-valued maps on non-unital C*-algebras. We also deduce some norm inequalities for these maps. Finally, we consider a noncommutative $L^2$-space equipped with the topology generated by a positive sesquilinear form and we construct a topologically transitive operator on this space. MSC 2020: 46K10, 47A07, 16D10, 37Bxx, 47G10. Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 06 Sep, 2024 Read the published version in Positivity → Version 1 posted Editorial decision: Accepted 12 Aug, 2024 Reviews received at journal 06 Aug, 2024 Reviewers agreed at journal 22 Jul, 2024 Reviewers agreed at journal 14 May, 2024 Reviewers agreed at journal 13 May, 2024 Reviewers invited by journal 11 May, 2024 Submission checks completed at journal 11 May, 2024 Editor assigned by journal 11 May, 2024 First submitted to journal 10 May, 2024 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-4402147","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":302477378,"identity":"3db196ef-31be-44bf-b009-266094436027","order_by":0,"name":"Giorgia Bellomonte","email":"data:image/png;base64,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","orcid":"","institution":"University of Palermo","correspondingAuthor":true,"submittingAuthor":false,"prefix":"","firstName":"Giorgia","middleName":"","lastName":"Bellomonte","suffix":""},{"id":302477382,"identity":"b3fac80c-ac9b-4049-b328-754a2c195cf9","order_by":1,"name":"Bogdan Djordjevic","email":"","orcid":"","institution":"Mathematical Institute of the Serbian Academy of Sciences and Arts","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Bogdan","middleName":"","lastName":"Djordjevic","suffix":""},{"id":302477384,"identity":"1589f312-db9f-40b8-9d35-3a858ab624da","order_by":2,"name":"Stefan Ivkovich","email":"","orcid":"","institution":"Mathematical Institute of the Serbian Academy of Sciences and Arts","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Stefan","middleName":"","lastName":"Ivkovich","suffix":""}],"badges":[],"createdAt":"2024-05-10 17:59:05","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-4402147/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-4402147/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s11117-024-01079-8","type":"published","date":"2024-09-06T15:56:57+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":64185661,"identity":"3c2a7da6-266a-493d-9fa3-ab35319cc1a4","added_by":"auto","created_at":"2024-09-09 16:20:15","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":324773,"visible":true,"origin":"","legend":"","description":"","filename":"BellomonteDjordjevicIvkovic.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4402147/v1_covered_0c8eb965-e5cf-431b-9653-c34680bc59a7.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"On representations and topological aspects of positive maps on non-unital quasi *- algebras","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"positivity","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"post","sideBox":"Learn more about [Positivity](http://link.springer.com/journal/11117)","snPcode":"11117","submissionUrl":"https://submission.nature.com/new-submission/11117/3","title":"Positivity","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"","lastPublishedDoi":"10.21203/rs.3.rs-4402147/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-4402147/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eIn this paper we provide a representation of a certain class of C*-valued positive sesquilinear and linear maps on non-unital quasi *- algebras, thus extending the results from [\\cite{GBSICT}] to the case of non-unital quasi *-algebras. Also, we illustrate our results on the concrete examples of non-unital Banach quasi *-algebras, such as the standard Hilbert module over a commutative C*-algebra, Schatten p-ideals and noncommutative $L^2$-spaces induced by a semifinite, nonfinite trace. As a consequence of our results, we obtain a representation of all bounded positive linear \u0026nbsp;C*-valued \u0026nbsp;maps on non-unital C*-algebras. We also deduce some norm inequalities for these maps. Finally, we consider a noncommutative $L^2$-space equipped with the topology generated by a positive sesquilinear form and we construct a topologically transitive operator on this space.\u003c/p\u003e\n\u003cp\u003eMSC 2020: 46K10, 47A07, 16D10, 37Bxx, 47G10.\u003c/p\u003e","manuscriptTitle":"On representations and topological aspects of positive maps on non-unital quasi *- algebras","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-05-21 17:12:46","doi":"10.21203/rs.3.rs-4402147/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Accepted","date":"2024-08-12T15:26:08+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2024-08-06T18:04:42+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"20765766070284658998144141262351800177","date":"2024-07-22T11:03:25+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"296987567918316165568635076190005975682","date":"2024-05-14T18:18:15+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"140286112801581322573340410226769424903","date":"2024-05-13T12:28:17+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2024-05-12T02:30:13+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2024-05-11T10:01:42+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2024-05-11T10:01:42+00:00","index":"","fulltext":""},{"type":"submitted","content":"Positivity","date":"2024-05-10T17:53:40+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"positivity","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"post","sideBox":"Learn more about [Positivity](http://link.springer.com/journal/11117)","snPcode":"11117","submissionUrl":"https://submission.nature.com/new-submission/11117/3","title":"Positivity","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"51795340-3c68-4f33-ae83-2f49cb9a8cc0","owner":[],"postedDate":"May 21st, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[],"tags":[],"updatedAt":"2024-09-09T16:09:42+00:00","versionOfRecord":{"articleIdentity":"rs-4402147","link":"https://doi.org/10.1007/s11117-024-01079-8","journal":{"identity":"positivity","isVorOnly":false,"title":"Positivity"},"publishedOn":"2024-09-06 15:56:57","publishedOnDateReadable":"September 6th, 2024"},"versionCreatedAt":"2024-05-21 17:12:46","video":"","vorDoi":"10.1007/s11117-024-01079-8","vorDoiUrl":"https://doi.org/10.1007/s11117-024-01079-8","workflowStages":[]},"version":"v1","identity":"rs-4402147","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-4402147","identity":"rs-4402147","version":["v1"]},"buildId":"7rjqhiLT3MXkJMwkYKINL","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}