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Relative bicentralizers and core commutants

Relative bicentralizers of inclusions with expectation and the commutant of the bottom algebra in the continuous core. The relative bicentralizer, its description in the Ocneanu ultrapower, its expectation as a limit of averages over almost centralizing unitaries; Connes' transition isomorphisms, the relative bicentralizer flow of Ando, Haagerup, Houdayer and Marrakchi, and the state-independent algebra generated by the bottom algebra and the relative bicentralizer; Marrakchi's binormal identity, proved through a tracial corner of the continuous core; approximate eigenoperators and Marrakchi's spectral shift maps; and OpenAI's proof (2026) that the two relative products of the bottom algebra and the relative bicentralizer differ exactly by the flow, through an iterated completely positive kernel, an estimate uniform in total modular energy and repeated random tensor tests. Consequences: the core fixed point theorem and a criterion for subalgebras with the same relative commutant in the core. Finally, finite-dimensional subfactors, state changes and the lifting of matrix units from the asymptotic centralizer, and the main theorem of OpenAI for bottom factors of type III_1: an approximately finite-dimensional subalgebra, invariant under the modular group of a suitable state, with the same relative commutant in the continuous core, built by an inductive construction after Ando, Haagerup, Houdayer and Marrakchi; and, for a single factor of type III_1, Haagerup's theorem that the faithful normal states with irreducible centralizer are norm dense. The last lesson treats semifinite bottom factors: Popa's hyperfinite subfactor with the same relative commutant for factors of type II_1, a matrix amplification with an almost periodic state for type II_infinity, and the resulting theorem for every semifinite bottom factor.

Eleven lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

  1. The relative bicentralizer
  2. Transition isomorphisms and the relative flow
  3. Binormal states and the relative bicentralizer
  4. Spectral shift maps
  5. The two relative products
  6. A uniform energy estimate
  7. Repeated tensor tests
  8. Core commutants
  9. Finite-dimensional subfactors and lifting
  10. An injective subalgebra with the same core commutant
  11. Semifinite bottom factors

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites