{ "id": "kadison-kaplansky-projections", "title": "A torsion-free counterexample to the Kadison–Kaplansky conjecture", "description": "Kadison and Kaplansky conjectured that the reduced group C*-algebra of a torsion-free group contains no projections other than 0 and 1. OpenAI constructed in 2026 a finitely generated torsion-free group with a projection whose canonical trace lies strictly between 0 and 1/2. The group is defined by the closed walks of a voltage cover of an infinite layered expander, the tower, with voltages in a torsion-free nilpotent group; a weighted adjacency matrix over its group ring has a separated spectral band of small trace, and compression after a free product gives the scalar projection. The tower is built from streams and gated permutations and pruned so that long paths are recovered from their words; the voltages realize a model operator assembled from Schwartz idempotents of twisted group algebras, with one random draw per edge. The course is an exposition of OpenAI's proof (2026).", "units": [ { "id": "the-conjecture-and-scalar-compression", "title": "The Kadison–Kaplansky conjecture and scalar compression", "reader": "the-conjecture-and-scalar-compression.html", "source": "src/the-conjecture-and-scalar-compression.md", "source_sha256": "5b923d4a4bef3e0d5f3803c89387ea2710fb7a67b6bf9d2c62bd6237d6989cf7", "status": "author_self_checked" }, { "id": "separated-bands-and-equivariant-traces", "title": "Separated bands and equivariant traces", "reader": "separated-bands-and-equivariant-traces.html", "source": "src/separated-bands-and-equivariant-traces.md", "source_sha256": "abf4aa9c6356fe66421207b2cbb29ff1ee349bda16c54fec905711ec7df9e0c0", "status": "author_self_checked" }, { "id": "streams-slots-and-slot-evaluations", "title": "Streams, slots and slot evaluations", "reader": "streams-slots-and-slot-evaluations.html", "source": "src/streams-slots-and-slot-evaluations.md", "source_sha256": "7571273c2192b13851cbabeb2dce3a0d8038ef24df529ec32af6e155adb96943", "status": "author_self_checked" }, { "id": "gated-permutations-and-the-two-towers", "title": "Gated permutations and the two towers", "reader": "gated-permutations-and-the-two-towers.html", "source": "src/gated-permutations-and-the-two-towers.md", "source_sha256": "86d000e47f010f3b2e58dc8f23be32819fdcb5b5415ab020833158f720e846c5", "status": "author_self_checked" }, { "id": "the-mean-incidence-and-its-gap", "title": "The mean incidence and its gap", "reader": "the-mean-incidence-and-its-gap.html", "source": "src/the-mean-incidence-and-its-gap.md", "source_sha256": "6792bd0c7565a4f611383c7284d332b9e2621b605c7602df1a989ba60abed956", "status": "author_self_checked" }, { "id": "expansion-of-the-realized-incidence", "title": "Expansion of the realized incidence", "reader": "expansion-of-the-realized-incidence.html", "source": "src/expansion-of-the-realized-incidence.md", "source_sha256": "4f19ee1b399a5cef7be0047973d5912788512fb35183bd07d99c38367d9ea57c", "status": "author_self_checked" }, { "id": "pruning-and-recovery-of-long-paths", "title": "Pruning and recovery of long paths", "reader": "pruning-and-recovery-of-long-paths.html", "source": "src/pruning-and-recovery-of-long-paths.md", "source_sha256": "d534b822015930d4e95c7bf0b45d1a4da6be527afa52abbaf6e6185040334c65", "status": "author_self_checked" }, { "id": "voltage-covers-and-embedded-copies", "title": "Voltage covers and embedded copies", "reader": "voltage-covers-and-embedded-copies.html", "source": "src/voltage-covers-and-embedded-copies.md", "source_sha256": "c569712fc2d9bd3781a5a6df194ef1d72be4c1a7218b74f7eb3a2da45bf4e6f9", "status": "author_self_checked" }, { "id": "thin-triangles-overlap-balls-and-the-residual-forest", "title": "Thin triangles, overlap balls and the residual forest", "reader": "thin-triangles-overlap-balls-and-the-residual-forest.html", "source": 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"b15cb7d7c1e0d8afaedf08f46edf9696db6b2bf0da83ab283f2de8467e2c2bd7", "status": "author_self_checked" }, { "id": "schwartz-idempotents-and-lattice-projections", "title": "Schwartz idempotents and lattice projections", "reader": "schwartz-idempotents-and-lattice-projections.html", "source": "src/schwartz-idempotents-and-lattice-projections.md", "source_sha256": "288393e04ee33934abe9b933c4ee045d31790521f9e5979414e6206771deb992", "status": "author_self_checked" }, { "id": "the-magnetic-factor-and-the-weighted-trace", "title": "The magnetic factor and the weighted trace", "reader": "the-magnetic-factor-and-the-weighted-trace.html", "source": "src/the-magnetic-factor-and-the-weighted-trace.md", "source_sha256": "3c827bbd3a78b1203187c70c33530c9bcae78af64aa6c143dff7127611f5f5d9", "status": "author_self_checked" }, { "id": "the-model-operator-and-finite-tests", "title": "The model operator and finite tests", "reader": "the-model-operator-and-finite-tests.html", "source": "src/the-model-operator-and-finite-tests.md", "source_sha256": "fbf85ccd645f389511ef0697039d1edf5b4f446c85a89899c1c9dfe5e27c0747", "status": "author_self_checked" }, { "id": "masks-lists-and-one-draw-per-edge", "title": "Masks, lists and one draw per edge", "reader": "masks-lists-and-one-draw-per-edge.html", "source": "src/masks-lists-and-one-draw-per-edge.md", "source_sha256": "a0d63facd50f72cd3e7dbf24af08a5ec85d447356ca44b8f2f292e3375a8d3fa", "status": "author_self_checked" }, { "id": "girth-and-centered-moments", "title": "Girth and centered moments", "reader": "girth-and-centered-moments.html", "source": "src/girth-and-centered-moments.md", "source_sha256": "81d00acd0803e23cb34f23b45e2602663585451d45444182708faff002fa8149", "status": "author_self_checked" }, { "id": "from-the-cover-band-to-the-cayley-projection", "title": "From the cover band to the Cayley projection", "reader": "from-the-cover-band-to-the-cayley-projection.html", "source": "src/from-the-cover-band-to-the-cayley-projection.md", "source_sha256": "514d6cf91b0090e127f2fb69123528248e8742250be3d3afd0be4d1a1a456c6a", "status": "author_self_checked" } ], "prerequisites": [ "programme course Foundations of von Neumann algebras: lessons C*-algebras, continuous functional calculus, automatic continuity and positive cones (Theorems 3.2 and 5.1, Corollary 5.4) and The spectral theorem for bounded self-adjoint operators (Proposition 5.2)", "programme course A projection of irrational trace and the Baum-Connes conjecture (irrational-trace-and-baum-connes): lessons Traces of projections and the Baum-Connes conjecture (Lemma 1.1, Section 3) and Graphical presentations and coned diagrams (Section 3)", "programme course Index theory of elliptic operators: lesson Hermite functions, tempered distributions and the Schwartz kernel theorem (Theorem 3.2, Proposition 2.2, Corollary 4.2)", "programme course The Bass trace conjecture for group rings (bass-trace-conjecture): lesson Idempotents and integral traces (Theorem 2.1), for context", "programme course Van Kampen diagrams (van-kampen-diagrams): lessons Plane maps (Propositions 1.2, 2.1, 2.2 and 2.3) and Van Kampen's lemma (Definition 1.1, Theorems 3.1, 4.1 and 5.1), which prove the statements (VK) and (MAP)" ], "status": "author_self_checked", "proof_providers": "providers.json", "provenance": "provenance.json", "licence": "CC0-1.0" }