{"id":"https://openalex.org/W1995307231","doi":"https://doi.org/10.4310/maa.2004.v11.n3.a5","title":"Operator-valued Free Entropy and Modular Frames","display_name":"Operator-valued Free Entropy and Modular Frames","publication_year":2004,"publication_date":"2004-01-01","ids":{"openalex":"https://openalex.org/W1995307231","doi":"https://doi.org/10.4310/maa.2004.v11.n3.a5","mag":"1995307231"},"language":"en","primary_location":{"id":"doi:10.4310/maa.2004.v11.n3.a5","doi":"https://doi.org/10.4310/maa.2004.v11.n3.a5","is_oa":true,"landing_page_url":"https://doi.org/10.4310/maa.2004.v11.n3.a5","pdf_url":"http://www.intlpress.com/site/pub/files/_fulltext/journals/maa/2004/0011/0003/MAA-2004-0011-0003-a005.pdf","source":{"id":"https://openalex.org/S2764760990","display_name":"Methods and Applications of Analysis","issn_l":"1073-2772","issn":["1073-2772","1945-0001"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["cwts-core","dongbi-c","jufo-1","norway-1","poland-20","russia-white-list-4"],"host_organization":"https://openalex.org/P4327904645","host_organization_name":"International Press of Boston","host_organization_lineage":["https://openalex.org/P4327904645"],"host_organization_lineage_names":["International Press of Boston"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Methods and Applications of Analysis","raw_type":"journal-article"},"volume":null,"type":"article","is_traditional":true,"indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"bronze","oa_url":"http://www.intlpress.com/site/pub/files/_fulltext/journals/maa/2004/0011/0003/MAA-2004-0011-0003-a005.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5102213362","display_name":"Xiaohong Cao","orcid":"https://orcid.org/0000-0002-9269-6679","observed_orcids":["https://orcid.org/0000-0002-9269-6679"]},"institutions":[{"id":"https://openalex.org/I20231570","display_name":"Peking University","ror":"https://ror.org/02v51f717","country_code":"CN","type":"education","lineage":["https://openalex.org/I20231570"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Xiaohong Cao","raw_affiliation_strings":["LMAM, School of Mathematical Science, Peking University, Beijing, 100871, P. R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"LMAM, School of Mathematical Science, Peking University, Beijing, 100871, P. R. China","institution_ids":["https://openalex.org/I20231570"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5108450827","display_name":"Maozheng Guo","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I20231570","display_name":"Peking University","ror":"https://ror.org/02v51f717","country_code":"CN","type":"education","lineage":["https://openalex.org/I20231570"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Maozheng Guo","raw_affiliation_strings":["LMAM, School of Mathematical Science, Peking University, Beijing, 100871, P. R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"LMAM, School of Mathematical Science, Peking University, Beijing, 100871, P. R. China","institution_ids":["https://openalex.org/I20231570"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5109878236","display_name":"Bin Meng","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Bin Meng","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5108450827"],"corresponding_institution_ids":["https://openalex.org/I20231570"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":2,"citation_normalized_percentile":{"value":0.0,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"11","issue":"3","first_page":"331","last_page":"344"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10899","display_name":"Advanced Operator Algebra Research","score":0.9965999722480774,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10899","display_name":"Advanced Operator Algebra Research","score":0.9965999722480774,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11716","display_name":"Random Matrices and Applications","score":0.0012000000569969416,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10884","display_name":"Holomorphic and Operator Theory","score":0.0005000000237487257,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/von-neumann-algebras","display_name":"von Neumann algebras","score":0.6169999837875366}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8866733312606812},{"id":"https://openalex.org/C101468663","wikidata":"https://www.wikidata.org/wiki/Q1620158","display_name":"Modular design","level":2,"score":0.5824639201164246},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.5410304069519043},{"id":"https://openalex.org/C106301342","wikidata":"https://www.wikidata.org/wiki/Q4117933","display_name":"Entropy (arrow of time)","level":2,"score":0.43819981813430786},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.3899341821670532},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.32685405015945435},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.07307115197181702},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","level":2,"score":0.0},{"id":"https://openalex.org/C55493867","wikidata":"https://www.wikidata.org/wiki/Q7094","display_name":"Biochemistry","level":1,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","level":1,"score":0.0},{"id":"https://openalex.org/C158448853","wikidata":"https://www.wikidata.org/wiki/Q425218","display_name":"Repressor","level":4,"score":0.0},{"id":"https://openalex.org/C86339819","wikidata":"https://www.wikidata.org/wiki/Q407384","display_name":"Transcription factor","level":3,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.4310/maa.2004.v11.n3.a5","doi":"https://doi.org/10.4310/maa.2004.v11.n3.a5","is_oa":true,"landing_page_url":"https://doi.org/10.4310/maa.2004.v11.n3.a5","pdf_url":"http://www.intlpress.com/site/pub/files/_fulltext/journals/maa/2004/0011/0003/MAA-2004-0011-0003-a005.pdf","source":{"id":"https://openalex.org/S2764760990","display_name":"Methods and Applications of Analysis","issn_l":"1073-2772","issn":["1073-2772","1945-0001"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["cwts-core","dongbi-c","jufo-1","norway-1","poland-20","russia-white-list-4"],"host_organization":"https://openalex.org/P4327904645","host_organization_name":"International Press of Boston","host_organization_lineage":["https://openalex.org/P4327904645"],"host_organization_lineage_names":["International Press of Boston"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Methods and Applications of Analysis","raw_type":"journal-article"},{"id":"pmh:oai:CULeuclid:euclid.maa/1147353057","doi":null,"is_oa":false,"landing_page_url":"http://projecteuclid.org/euclid.maa/1147353057","pdf_url":null,"source":{"id":"https://openalex.org/S4306400787","display_name":"Project Euclid (Cornell University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.4310/maa.2004.v11.n3.a5","doi":"https://doi.org/10.4310/maa.2004.v11.n3.a5","is_oa":true,"landing_page_url":"https://doi.org/10.4310/maa.2004.v11.n3.a5","pdf_url":"http://www.intlpress.com/site/pub/files/_fulltext/journals/maa/2004/0011/0003/MAA-2004-0011-0003-a005.pdf","source":{"id":"https://openalex.org/S2764760990","display_name":"Methods and Applications of Analysis","issn_l":"1073-2772","issn":["1073-2772","1945-0001"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["cwts-core","dongbi-c","jufo-1","norway-1","poland-20","russia-white-list-4"],"host_organization":"https://openalex.org/P4327904645","host_organization_name":"International Press of Boston","host_organization_lineage":["https://openalex.org/P4327904645"],"host_organization_lineage_names":["International Press of Boston"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Methods and Applications of Analysis","raw_type":"journal-article"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W1995307231.pdf","grobid_xml":"https://content.openalex.org/works/W1995307231.grobid-xml"},"referenced_works_count":18,"referenced_works":["https://openalex.org/W1568450341","https://openalex.org/W1590503522","https://openalex.org/W1666093761","https://openalex.org/W1989292836","https://openalex.org/W2002490441","https://openalex.org/W2006827575","https://openalex.org/W2026938829","https://openalex.org/W2043256123","https://openalex.org/W2044442377","https://openalex.org/W2075987172","https://openalex.org/W2099111195","https://openalex.org/W2166361939","https://openalex.org/W2316317025","https://openalex.org/W4389255257","https://openalex.org/W6602126913","https://openalex.org/W6629048134","https://openalex.org/W6637642161","https://openalex.org/W6688798157"],"related_works":["https://openalex.org/W1570615215","https://openalex.org/W2044314303","https://openalex.org/W2056669594","https://openalex.org/W2150130074","https://openalex.org/W7100308706","https://openalex.org/W7097546223","https://openalex.org/W7100033776","https://openalex.org/W2950995131","https://openalex.org/W2067785318","https://openalex.org/W1031760853"],"abstract_inverted_index":{"We":[0],"introduce":[1],"the":[2,89,97,100,108,120],"operator-valued":[3],"relative":[4,56],"free":[5,57,74,101,121],"entropy":[6,58,75,102,122],"\u03c7":[7],"*":[8,60,125],"B":[9,49],"(X":[10],"1":[11,31],",":[12,15,19,32,35,39,50],"X":[13,30,33],"2":[14,34],"\u2022":[16,17,18,36,37,38],"Xn":[20,40],":":[21],"B)":[22],"of":[23,26,76,103,110,119],"a":[24,42,64,70,77,104,111],"family":[25],"self-adjoint":[27],"random":[28],"variables":[29],"in":[41,63],"B-valued":[43],"noncommutative":[44],"probability":[45,67],"space":[46,68],"(A,":[47],"E":[48],"B).This":[51],"notion":[52],"extends":[53],"D.":[54],"Voiculescu's":[55],"\u03a6":[59],"which":[61],"defined":[62],"tracial":[65],"W*-noncommutative":[66],"to":[69],"more":[71],"general":[72],"context.The":[73],"semicircular":[78,105],"variable":[79,106],"with":[80],"conditional":[81,112],"expectation":[82],"covariance":[83],"can":[84],"be":[85],"computed":[86],"by":[87],"using":[88],"modular":[90],"frames":[91],"and":[92,107],"then":[93],"we":[94,115],"point":[95],"out":[96],"relation":[98],"between":[99],"index":[109],"expectation.At":[113],"last,":[114],"obtain":[116],"an":[117],"estimate":[118],"dimension":[123],"\u03b4":[124],"B,\u03c4":[126],".":[127]},"counts_by_year":[],"updated_date":"2026-10-30T12:39:25.321706","created_date":"2016-06-24T00:00:00"}
