{"paper":{"title":"Continuous categories of endomorphisms associated with $G$-kernels","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Continuous categories of endomorphisms of type III factors arise from G-kernels on compact second countable groups.","cross_cats":["math-ph","math.CT","math.FA","math.MP","math.QA"],"primary_cat":"math.OA","authors_text":"Marcel Bischoff, Pradyut Karmakar","submitted_at":"2026-05-17T15:59:19Z","abstract_excerpt":"We generalize the construction of tensor categories of endomorphisms of a type III factor $M$ associated with a $G$-kernel, from the case of a discrete group $G$ to that of a compact second countable group. Our approach is based on the construction of a unitary tensor functor from a category of $C(G)$-modules to the category of endomorphisms of $M$. This functor maps a $C(G)$-module, realized as the space of square-integrable functions on a measure space, to a continuous family of endomorphisms of $M$. The resulting structure is a continuous category of endomorphisms, which provides a new fram"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We generalize the construction of tensor categories of endomorphisms of a type III factor M associated with a G-kernel, from the case of a discrete group G to that of a compact second countable group.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The approach assumes the existence of a unitary tensor functor from the category of C(G)-modules, realized as square-integrable functions on a measure space, to the category of endomorphisms of M that produces a continuous family for compact second countable G.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Generalizes discrete G-kernel endomorphism categories to compact groups using a unitary tensor functor from C(G)-modules to produce continuous families of endomorphisms.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Continuous categories of endomorphisms of type III factors arise from G-kernels on compact second countable groups.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"15895d5998190ffc7a1ae65912caa38c81dc97cefbe9cdc4ae98dcddc6ba1fe6"},"source":{"id":"2605.17514","kind":"arxiv","version":1},"verdict":{"id":"f5090b2d-4934-49ac-aa89-abe1a77501cb","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T22:32:35.186717Z","strongest_claim":"We generalize the construction of tensor categories of endomorphisms of a type III factor M associated with a G-kernel, from the case of a discrete group G to that of a compact second countable group.","one_line_summary":"Generalizes discrete G-kernel endomorphism categories to compact groups using a unitary tensor functor from C(G)-modules to produce continuous families of endomorphisms.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The approach assumes the existence of a unitary tensor functor from the category of C(G)-modules, realized as square-integrable functions on a measure space, to the category of endomorphisms of M that produces a continuous family for compact second countable G.","pith_extraction_headline":"Continuous categories of endomorphisms of type III factors arise from G-kernels on compact second countable groups."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.17514/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T23:01:19.516454Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T22:41:16.829266Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T21:41:57.652026Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T21:33:23.628686Z","status":"skipped","version":"1.0.0","findings_count":0}],"snapshot_sha256":"503c42da1fc71602a218820120228b3705075e042083e0cae8506140dde5246f"},"references":{"count":4,"sample":[{"doi":"","year":1969,"title":"MR3308880 24 [DHR69] S. Doplicher, R. Haag, and J. E. Roberts,Fields, observables and gauge transformations II, Comm. Math. Phys.15(1969), 173–200. [EGNO15] P. Etingof, S. Gelaki, D. Nikshych, and V. ","work_id":"8ea9ed7e-98f9-4397-9346-f37d3696c2d9","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1985,"title":"MR3242743 [GLR85] P. Ghez, R. Lima, and J. E. Roberts,W∗-categories, Pacific J. Math.120(1985), no. 1, 79–109. MR808930 [Haa75] U. Haagerup,The standard form of von Neumann algebras, Math. Scand.37(19","work_id":"39d4b4a0-fd8e-4a37-9baa-6eb6752dd044","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2015,"title":"A Cuntz algebra approach to the classification of near-group categories","work_id":"997da528-a91e-498a-9d73-4767462a2e74","ref_index":3,"cited_arxiv_id":"1512.04288","is_internal_anchor":true},{"doi":"","year":1980,"title":"[Sut80] C. E. Sutherland,Cohomology and extensions of von Neumann algebras. II, Publ. Res. Inst. Math. 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