{"paper":{"title":"Kadison's problem for trace-vector orthonormal bases in $\\mathrm{II}_1$ factors with separable predual","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Diffuse finite von Neumann algebras with separable L2 admit orthonormal bases of self-adjoint unitaries.","cross_cats":[],"primary_cat":"math.OA","authors_text":"Quanyu Tang, Teng Zhang, Yixin He","submitted_at":"2026-05-14T16:07:06Z","abstract_excerpt":"In 1967, Kadison asked ``does every type $\\mathrm{II}_1$ factor have an orthonormal (with respect to the trace) basis consisting of unitaries?'' Using a noncommutative Lyapunov theorem of Akemann and Weaver, we show that finite dimensional orthogonality constraints can be realized by projections, and hence by symmetries. Iterating this construction, we prove that if $M$ is a diffuse finite von Neumann algebra with faithful normal tracial state $\\tau$ and $L^2(M,\\tau)$ is separable, then $L^2(M,\\tau)$ admits an orthonormal basis consisting of self-adjoint unitaries in $M$. Consequently, we affi"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"if M is a diffuse finite von Neumann algebra with faithful normal tracial state τ and L²(M,τ) is separable, then L²(M,τ) admits an orthonormal basis consisting of self-adjoint unitaries in M. Consequently, we affirm the separable case of the Kadison problem.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The Akemann-Weaver noncommutative Lyapunov theorem can be applied to the specific finite-dimensional orthogonality constraints that arise at each iterative step, and the reduced algebra after removing the span of the chosen symmetries remains diffuse so that the process can continue indefinitely.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Every diffuse finite von Neumann algebra with separable L2 space has an orthonormal basis of self-adjoint unitaries with respect to the trace.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Diffuse finite von Neumann algebras with separable L2 admit orthonormal bases of self-adjoint unitaries.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"219571532c14a7e342b0ab33d71e4d23c3f487d8268a1b0763b80c3ad5739f87"},"source":{"id":"2605.15006","kind":"arxiv","version":1},"verdict":{"id":"bd4aea48-c45c-4623-9a09-022f89a8e12c","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T02:43:16.933447Z","strongest_claim":"if M is a diffuse finite von Neumann algebra with faithful normal tracial state τ and L²(M,τ) is separable, then L²(M,τ) admits an orthonormal basis consisting of self-adjoint unitaries in M. Consequently, we affirm the separable case of the Kadison problem.","one_line_summary":"Every diffuse finite von Neumann algebra with separable L2 space has an orthonormal basis of self-adjoint unitaries with respect to the trace.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The Akemann-Weaver noncommutative Lyapunov theorem can be applied to the specific finite-dimensional orthogonality constraints that arise at each iterative step, and the reduced algebra after removing the span of the chosen symmetries remains diffuse so that the process can continue indefinitely.","pith_extraction_headline":"Diffuse finite von Neumann algebras with separable L2 admit orthonormal bases of self-adjoint unitaries."},"references":{"count":7,"sample":[{"doi":"","year":2003,"title":"C. A. Akemann and N. Weaver, Automatic convexity, J. Convex Anal. 10 (2003), no. 1, 275--284. https://www.heldermann-verlag.de/jca/jca10/jca0339.pdf","work_id":"9f49d409-ce19-4541-96cd-5351dd77ad4f","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1090/s0002-9947-1973-0326405-3","year":1973,"title":"Ching, Free products of von Neumann algebras, Trans","work_id":"65966ac0-d81b-4a44-a762-379a3de60848","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1987,"title":"M. Choda, Shifts on the hyperfinite II_1 -factor , J. Operator Theory 17 (1987), no. 2, 223--235. https://www.jstor.org/stable/24714840","work_id":"2e00eb6c-1e4f-4aed-b86c-8a9bcba4b5fe","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.4171/dm/941","year":2023,"title":"D. De and K. Mukherjee, On the existence of uniformly bounded self-adjoint bases in GNS spaces, Doc. Math. 28 (2023), 1381--1392. doi:10.4171/DM/941","work_id":"a45e6064-4f92-44ed-8e1c-7e3248572269","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1007/s10114-003-0279-x","year":1967,"title":"L. M. Ge, On ``Problems on von Neumann Algebras by R. Kadison, 1967'', Acta Math. Sin. (Engl. Ser.) 19 (2003), no. 3, 619--624. doi:10.1007/s10114-003-0279-x","work_id":"203351ab-ec62-45fb-af4a-fc8687e61bbe","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":7,"snapshot_sha256":"b753693044e08a70e3c923b86bc04b87d744a36c253ca7e23d26e93a8fa25db4","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"803c5cbcd6d63e24388562e7c181e68943cd023f61065f907bdc2815430c4155"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}