{"paper":{"title":"Localizing the Elliott conjecture at strongly self-absorbing C*-algebras","arxiv_id":"0708.0283","primary_cat":"math.OA","submitted_at":"2007-08-02T07:04:22+00:00","version_compiled":"v3"},"notice":"Every quoted sentence is the authors' text from the arXiv listing abstract. Labels and probabilities are the outputs of classifier-1.13-20260917 on the question bank abstract-p1-20260923, exactly as returned. No sentence in this record was written by a language model.","schema":"pith.compiled-record/v0","status":"compiled","abstract":{"source":"arXiv listing abstract v3","sentences":[{"i":0,"p":0.96,"id":"s0","own":true,"role":"contribution","text":"We formally introduce the concept of localizing the Elliott conjecture at a given strongly self-absorbing C*-algebra $D$; we also explain how the known classification theorems for nuclear C*-algebras fit into this concept.","hedge":{"p":0.99,"label":"asserted"},"own_p":0.89,"main_claim_p":0.06,"quantitative":false,"quantitative_p":0.03,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":0.04,"limitation":0.0,"future_work":0.0,"contribution":0.96,"interpretation":0.0}},{"i":1,"p":0.91,"id":"s1","own":true,"role":"result","text":"As a new result in this direction, we employ recent results of Lin to show that (under a mild K-theoretic condition) the class of separable, unital, simple C*-algebras with locally finite decomposition rank and UCT, and for which projections separate traces, satisfies the Elliott conjecture localized at the Jiang-Su algebra Z.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.93,"main_claim_p":0.28,"quantitative":false,"quantitative_p":0.08,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.91,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.09,"interpretation":0.0}},{"i":2,"p":0.91,"id":"s2","own":true,"role":"future_work","text":"Our main result is formulated in a more general way; this allows us to outline a strategy to possibly remove the trace space condition as well as the K-theory restriction entirely.","hedge":{"p":0.84,"label":"hedged_possibility"},"own_p":0.85,"main_claim_p":0.22,"quantitative":false,"quantitative_p":0.06,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.02,"background":0.0,"limitation":0.0,"future_work":0.91,"contribution":0.03,"interpretation":0.04}},{"i":3,"p":0.99,"id":"s3","own":true,"role":"interpretation","text":"When regarding both our result and the recent classification theorem of Elliott, Gong and Li as generalizations of the real rank zero case, the two approaches are perpendicular in a certain sense.","hedge":{"p":0.98,"label":"asserted"},"own_p":0.59,"main_claim_p":0.04,"quantitative":false,"quantitative_p":0.34,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.01,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.99}},{"i":4,"p":0.86,"id":"s4","own":true,"role":"future_work","text":"The strategy to attack the general case aims at combining these two approaches.","hedge":{"p":0.79,"label":"asserted"},"own_p":0.52,"main_claim_p":0.05,"quantitative":false,"quantitative_p":0.05,"role_probabilities":{"gap":0.0,"method":0.09,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.86,"contribution":0.02,"interpretation":0.03}},{"i":5,"p":0.76,"id":"s5","own":true,"role":"result","text":"Our classification theorem covers simple ASH algebras for which projections separate traces (and the K-groups of which have finitely generated torsion part); it does, however, not at all depend on an inductive limit structure.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.92,"main_claim_p":0.18,"quantitative":false,"quantitative_p":0.15,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.76,"background":0.0,"limitation":0.15,"future_work":0.0,"contribution":0.06,"interpretation":0.03}},{"i":6,"p":0.42,"id":"s6","own":true,"role":"result","text":"Also, in the monotracial case it does not rely on the existence or absence of projections in any way.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.84,"main_claim_p":0.05,"quantitative":false,"quantitative_p":0.06,"role_probabilities":{"gap":0.0,"method":0.01,"result":0.42,"background":0.0,"limitation":0.06,"future_work":0.0,"contribution":0.26,"interpretation":0.25}},{"i":7,"p":0.39,"id":"s7","own":true,"role":"result","text":"In fact, it is the first such result which, in a natural way, covers all known unital, separable, simple, nuclear and stably finite C*-algebras of real rank zero (the K-groups of which have finitely generated torsion part) as well as the (projectionless) Jiang-Su algebra itself.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.91,"main_claim_p":0.23,"quantitative":false,"quantitative_p":0.39,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.39,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.28,"interpretation":0.33}}]},"compiled":{"main_claim":{"p":0.47,"text":"As a new result in this direction, we employ recent results of Lin to show that (under a mild K-theoretic condition) the class of separable, unital, simple C*-algebras with locally finite decomposition rank and UCT, and for which projections separate traces, satisfies the Elliott conjecture localized at the Jiang-Su algebra Z.","location":"abstract s1","hedge_strength":{"p":1.0,"label":"asserted"}},"provenance":{"route":{"via":"OpenRouter","model":"classifier-1.13","endpoint":"https://openrouter.ai/api/v1/systemone","provider":"TypeSafe"},"packing":"one request per paper","requests":1,"input_tokens":6599,"cost_usd_reported":0.000277},"compiled_at":"2026-09-23","bank_version":"abstract-p1-20260923","model_version":"classifier-1.13-20260917","abstract_states":{"claims_first":0.89,"states_limitation":0.93,"human_participants":0.01,"controlled_comparison":0.03,"randomized_assignment":0.01,"states_quantitative_result":0.05,"compares_with_prior_results":0.97,"mentions_code_or_data_availability":0.01},"evidential_basis":{"p":0.99,"value":"proof or derivation","probabilities":{"argument":0.0,"literature":0.01,"proof_or_derivation":0.99,"analysis_of_existing_data":0.0,"observation_or_experiment":0.0,"simulation_or_computation":0.0}},"contribution_type":{"p":0.99,"value":"theoretical result","probabilities":{"review":0.0,"method_or_tool":0.01,"empirical_study":0.0,"simulation_study":0.0,"theoretical_result":0.99,"dataset_or_benchmark":0.0,"position_or_commentary":0.0}},"contribution_statement":{"p":0.99,"text":"As a new result in this direction, we employ recent results of Lin to show that (under a mild K-theoretic condition) the class of separable, unital, simple C*-algebras with locally finite decomposition rank and UCT, and for which projections separate traces, satisfies the Elliott conjecture localized at the Jiang-Su algebra Z.","location":"abstract s1"}},"coverage":{"tier":"abstract","levels_run":["P abstract"],"not_examined":["full text","figures and tables","references and citing sentences","relations to other papers"],"sources_read":["arXiv listing abstract v3"],"levels_not_run":["S sections","T body sentences","C citations","R cross-paper relations"],"sentences_examined":8,"sentences_in_abstract":8},"measured_accuracy":{"url":"https://pith.science/accuracy","measured_on":"2026-10-02","method":"two independent model families labelled a fresh random sample blind; 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