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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":true,"formal_links_present":true},"canonical_record":{"source":{"id":"2605.15006","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2026-05-14T16:07:06Z","cross_cats_sorted":[],"title_canon_sha256":"ec8dce15a4e8008fbf3b067baee6ad1a1cbd8e99e1d847d85939c7c30e9e3ac4","abstract_canon_sha256":"69dc2bd519b7747907da1a30cb6dd6a56da92abb8be550b26203fc75fb6d3f8a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:38:54.867406Z","signature_b64":"FjCGoZPQzOh7yisOL3Ra0ly1VnIs07u/1dmbfqElDxt3m6AvpQRBi54OQrAA0WfRsqwa6NhnwytcBbBukfbxCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3569cd46221096b79c81a8a3153c4869e95527f1c952e44c9337e1ade551b94f","last_reissued_at":"2026-05-17T23:38:54.866716Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:38:54.866716Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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