{"id":"https://openalex.org/W7220712709","doi":"https://doi.org/10.48550/arxiv.2610.06771","title":"Quantum integrals (With an Appendix on the solution to Haagerup's weight problem)","display_name":"Quantum integrals (With an Appendix on the solution to Haagerup's weight problem)","publication_year":2026,"publication_date":"2026-10-05","ids":{"openalex":"https://openalex.org/W7220712709","doi":"https://doi.org/10.48550/arxiv.2610.06771"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.06771","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.06771","pdf_url":"https://arxiv.org/pdf/2610.06771","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"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":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2610.06771","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5015708217","display_name":"David P. Blecher","orcid":"https://orcid.org/0000-0003-2093-2289","observed_orcids":["https://orcid.org/0000-0003-2093-2289"]},"institutions":[{"id":"https://openalex.org/I44461941","display_name":"University of Houston","ror":"https://ror.org/048sx0r50","country_code":"US","type":"education","lineage":["https://openalex.org/I44461941"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Blecher, David P.","raw_affiliation_strings":["Department of Mathematics , University of Houston , Houston , TX 77204-3008 , USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , University of Houston , Houston , TX 77204-3008 , USA","institution_ids":["https://openalex.org/I44461941"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5008129827","display_name":"Louis E. Labuschagne","orcid":"https://orcid.org/0000-0002-0309-4765","observed_orcids":["https://orcid.org/0000-0002-0309-4765"]},"institutions":[{"id":"https://openalex.org/I27765905","display_name":"North-West University","ror":"https://ror.org/010f1sq29","country_code":"ZA","type":"education","lineage":["https://openalex.org/I27765905"]},{"id":"https://openalex.org/I44461941","display_name":"University of Houston","ror":"https://ror.org/048sx0r50","country_code":"US","type":"education","lineage":["https://openalex.org/I44461941"]}],"countries":["US","ZA"],"is_corresponding":false,"raw_author_name":"Labuschagne, Louis E.","raw_affiliation_strings":["Department of Mathematics , University of Houston , Houston , TX 77204-3008 , USA","Internal Box 209 , School of Comp. , Stat. & Math. Sci. , NWU , Pvt. Bag X6001 , 2520 Potchefstroom , South Africa"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , University of Houston , Houston , TX 77204-3008 , USA","institution_ids":["https://openalex.org/I44461941"]},{"raw_affiliation_string":"Internal Box 209 , School of Comp. , Stat. & Math. Sci. , NWU , Pvt. Bag X6001 , 2520 Potchefstroom , South Africa","institution_ids":["https://openalex.org/I27765905"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":3,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10899","display_name":"Advanced Operator Algebra Research","score":0.9990000128746033,"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":0.9990000128746033,"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/T11545","display_name":"Advanced Banach Space Theory","score":0.000699999975040555,"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/T10884","display_name":"Holomorphic and Operator Theory","score":0.0,"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":0.9660000205039978},{"id":"https://openalex.org/keywords/direct-integrals","display_name":"direct integrals","score":0.6230000257492065},{"id":"https://openalex.org/keywords/gleason-s-theorem","display_name":"Gleason's theorem","score":0.8679999709129333},{"id":"https://openalex.org/keywords/noncommutative-integration","display_name":"noncommutative integration","score":0.8009999990463257}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.753000020980835},{"id":"https://openalex.org/C153408630","wikidata":"https://www.wikidata.org/wiki/Q1970172","display_name":"Von Neumann algebra","level":3,"score":0.7527999877929688},{"id":"https://openalex.org/C80469333","wikidata":"https://www.wikidata.org/wiki/Q189088","display_name":"Von Neumann architecture","level":2,"score":0.73580002784729},{"id":"https://openalex.org/C68797384","wikidata":"https://www.wikidata.org/wiki/Q2746792","display_name":"Noncommutative geometry","level":2,"score":0.5756000280380249},{"id":"https://openalex.org/C57493831","wikidata":"https://www.wikidata.org/wiki/Q3134666","display_name":"Projection (relational algebra)","level":2,"score":0.5288000106811523},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.4912000000476837},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.4699999988079071},{"id":"https://openalex.org/C2780009758","wikidata":"https://www.wikidata.org/wiki/Q6804172","display_name":"Measure (data warehouse)","level":2,"score":0.4341999888420105},{"id":"https://openalex.org/C21261745","wikidata":"https://www.wikidata.org/wiki/Q7941491","display_name":"Von Neumann's theorem","level":4,"score":0.42340001463890076},{"id":"https://openalex.org/C48103436","wikidata":"https://www.wikidata.org/wiki/Q599031","display_name":"State (computer science)","level":2,"score":0.3894999921321869},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.3871999979019165},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.3822999894618988},{"id":"https://openalex.org/C2778029271","wikidata":"https://www.wikidata.org/wiki/Q5421931","display_name":"Extension (predicate logic)","level":2,"score":0.3806000053882599},{"id":"https://openalex.org/C27142425","wikidata":"https://www.wikidata.org/wiki/Q1139334","display_name":"Almost everywhere","level":2,"score":0.3767000138759613},{"id":"https://openalex.org/C84114770","wikidata":"https://www.wikidata.org/wiki/Q46344","display_name":"Quantum","level":2,"score":0.36010000109672546},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.35679998993873596},{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.35019999742507935},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.3425999879837036},{"id":"https://openalex.org/C54512446","wikidata":"https://www.wikidata.org/wiki/Q7818977","display_name":"Tomita\u2013Takesaki theory","level":5,"score":0.3255999982357025},{"id":"https://openalex.org/C144133476","wikidata":"https://www.wikidata.org/wiki/Q4666687","display_name":"Abelian von Neumann algebra","level":5,"score":0.322299987077713},{"id":"https://openalex.org/C163681178","wikidata":"https://www.wikidata.org/wiki/Q1149279","display_name":"Neumann boundary condition","level":3,"score":0.3174000084400177},{"id":"https://openalex.org/C137343772","wikidata":"https://www.wikidata.org/wiki/Q584910","display_name":"Neumann series","level":2,"score":0.30079999566078186},{"id":"https://openalex.org/C98184364","wikidata":"https://www.wikidata.org/wiki/Q1780131","display_name":"Argument (complex analysis)","level":2,"score":0.29910001158714294},{"id":"https://openalex.org/C138081561","wikidata":"https://www.wikidata.org/wiki/Q4688920","display_name":"Affiliated operator","level":5,"score":0.2953000068664551},{"id":"https://openalex.org/C118365987","wikidata":"https://www.wikidata.org/wiki/Q273328","display_name":"Integration by parts","level":2,"score":0.2867000102996826},{"id":"https://openalex.org/C70120467","wikidata":"https://www.wikidata.org/wiki/Q515893","display_name":"Free probability","level":2,"score":0.2833000123500824},{"id":"https://openalex.org/C36503486","wikidata":"https://www.wikidata.org/wiki/Q11235244","display_name":"Domain (mathematical analysis)","level":2,"score":0.2802000045776367},{"id":"https://openalex.org/C21031990","wikidata":"https://www.wikidata.org/wiki/Q355020","display_name":"Probability measure","level":2,"score":0.2718000113964081},{"id":"https://openalex.org/C183778304","wikidata":"https://www.wikidata.org/wiki/Q165474","display_name":"Commutative property","level":2,"score":0.27140000462532043},{"id":"https://openalex.org/C110729354","wikidata":"https://www.wikidata.org/wiki/Q185478","display_name":"Countable set","level":2,"score":0.2694999873638153},{"id":"https://openalex.org/C38487692","wikidata":"https://www.wikidata.org/wiki/Q1067156","display_name":"Dominated convergence theorem","level":5,"score":0.2655999958515167},{"id":"https://openalex.org/C114852677","wikidata":"https://www.wikidata.org/wiki/Q899044","display_name":"Fock space","level":2,"score":0.2655999958515167},{"id":"https://openalex.org/C64775481","wikidata":"https://www.wikidata.org/wiki/Q3798613","display_name":"Quantum mutual information","level":5,"score":0.26330000162124634},{"id":"https://openalex.org/C38349280","wikidata":"https://www.wikidata.org/wiki/Q1434290","display_name":"Flow (mathematics)","level":2,"score":0.25929999351501465},{"id":"https://openalex.org/C40422974","wikidata":"https://www.wikidata.org/wiki/Q741248","display_name":"Assertion","level":2,"score":0.25859999656677246},{"id":"https://openalex.org/C61039578","wikidata":"https://www.wikidata.org/wiki/Q604279","display_name":"Dirac (video compression format)","level":3,"score":0.25290000438690186},{"id":"https://openalex.org/C15706264","wikidata":"https://www.wikidata.org/wiki/Q230883","display_name":"Quantum state","level":3,"score":0.25060001015663147}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.06771","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.06771","pdf_url":"https://arxiv.org/pdf/2610.06771","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"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":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2610.06771","doi":"https://doi.org/10.48550/arxiv.2610.06771","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.06771","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"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":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2610.06771","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.06771","pdf_url":"https://arxiv.org/pdf/2610.06771","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"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":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[{"id":"https://openalex.org/G785469004","display_name":"Noncommutative Analysis with Applications to Quantum Information Theory","funder_award_id":"2154903","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7220712709.pdf","grobid_xml":"https://content.openalex.org/works/W7220712709.grobid-xml"},"referenced_works_count":18,"referenced_works":["https://openalex.org/W630175279","https://openalex.org/W1524878283","https://openalex.org/W1619459494","https://openalex.org/W2028498782","https://openalex.org/W2038231956","https://openalex.org/W2038383557","https://openalex.org/W2043698098","https://openalex.org/W2057558068","https://openalex.org/W2068887873","https://openalex.org/W2094947590","https://openalex.org/W2530602005","https://openalex.org/W2935921334","https://openalex.org/W2938045018","https://openalex.org/W3123677122","https://openalex.org/W4200515354","https://openalex.org/W4247375051","https://openalex.org/W4414938658","https://openalex.org/W7204268182"],"related_works":["https://openalex.org/W6995971239","https://openalex.org/W7221812998","https://openalex.org/W2016293317","https://openalex.org/W4412645902","https://openalex.org/W1986593152","https://openalex.org/W4236329051","https://openalex.org/W4243611101","https://openalex.org/W2055152402","https://openalex.org/W1517211899","https://openalex.org/W7095754278"],"abstract_inverted_index":{"We":[0,7,96,131],"solve":[1,9],"two":[2],"long":[3],"outstanding":[4],"problems":[5],"here.":[6],"first":[8,16,194],"in":[10,189],"ZFC":[11],"a":[12,24,27,65,84,104,223],"problem":[13],"that":[14,196],"the":[15,54,79,123,193],"author":[17,195],"and":[18,38,135,141,162,180,198],"Weaver":[19],"posed":[20],"ten":[21],"years":[22],"ago:":[23],"state":[25],"on":[26,34,89,115],"von":[28,93,174,211],"Neumann":[29,94,175,212],"algebra":[30,124],"is":[31,42,231],"countably":[32,111],"additive":[33,112],"orthogonal":[35],"projections":[36,116],"if":[37,40],"only":[39],"it":[41],"sequentially":[43,47,118],"weak*":[44,119],"continuous,":[45],"equivalently,":[46],"normal.":[48],"Such":[49],"states":[50,88,121],"may":[51,219],"be":[52,220],"considered":[53],"quantum":[55],"(countably":[56],"additive)":[57],"probability":[58,113],"measures,":[59],"or":[60],"rather":[61],"their":[62,207],"extension":[63],"to":[64,117,150,159,187,209,222],"noncommutative":[66],"integral.":[67],"The":[68,177],"principal":[69],"tools":[70,179],"are":[71,185,204],"known":[72],"ideas":[73],"from":[74,78],"direct":[75,129],"integral":[76],"theory":[77],"1970s.":[80],"Indeed":[81],"we":[82,102,155],"prove":[83,188],"disintegration":[85],"theorem":[86,172],"for":[87,139,173],"products":[90],"of":[91,107,146,168,201,229],"sigma-finite":[92],"algebras.":[95,176],"give":[97,103,142],"several":[98,143],"applications.":[99],"For":[100],"example":[101],"new":[105],"variant":[106],"Gleason":[108],"theorem,":[109],"extending":[110],"measures":[114],"continuous":[120],"when":[122],"has":[125],"no":[126],"type":[127],"$I_2$":[128],"summand.":[130],"also":[132],"establish":[133],"normality":[134,137,197,200,228],"sequential":[136,199],"criteria":[138],"weights,":[140],"other":[144],"applications":[145,158],"our":[147],"main":[148],"result":[149],"weights.":[151],"In":[152],"forthcoming":[153],"work":[154],"consider":[156],"many":[157],"`quantum":[160],"measure":[161],"integration":[163],"theory',":[164],"such":[165],"as":[166],"variants":[167],"Lebesgue's":[169],"dominated":[170],"convergence":[171],"analytic":[178],"product-quotient":[181],"construction":[182],"developed":[183],"here":[184],"used":[186],"an":[190],"Appendix":[191],"by":[192,206],"arbitrary":[202],"weights":[203,230],"determined":[205],"restrictions":[208],"abelian":[210],"subalgebras,":[213],"resolving":[214],"Haagerup's":[215],"Problem":[216],"1.11.":[217],"(This":[218],"moved":[221],"companion":[224],"paper.)":[225],"That":[226],"is,":[227],"commutatively":[232],"determined.":[233]},"counts_by_year":[{"year":2026,"cited_by_count":3}],"updated_date":"2026-11-08T12:12:28.789561","created_date":"2026-10-07T00:00:00"}