OAI.Kadison.PartialIsometryOrder.maximal_family_spans_of_full_support: intermediate lemma from OpenAI, Kadison's similarity theorem through uniform derivation estimates (result family 288)

Proved

OAI.Kadison.PartialIsometryOrder.maximal_family_spans_of_full_support

by dbenbenn · Oct 28, 2026 · Mathlib 0df444a (Lean v4.33.1)

openai-math

An intermediate lemma of the split of OpenAI's proof of OAI.Kadison.uniform_commutator, carved from OpenAI's openai/math development for this platform. The statement and its proof are OpenAI's (OpenAI, Kadison's similarity theorem through uniform derivation estimates (result family 288), OpenAI Math Release, September 23, 2026, result family 288, github.com/openai/math, Apache License 2.0), ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.Kadison.PartialIsometryOrder, with open scoped InnerProductSpace ComplexOrder Topology):

theorem maximal_family_spans_of_full_support
    (hsupport : (FactorCorner.supportSpace M e).starProjection=1)
    (heM : e∈M) (hfM : f∈M) (he : IsStarProjection e) (hf : IsStarProjection f)
    (S : Set (Link M e f)) (hS : Maximal (OrthogonalLinks M e f) S) :
    (familySpace M e f S).starProjection=f

Definitions. The statement is written with OAI.Kadison.PartialIsometryOrder.Link, OAI.Kadison.PartialIsometryOrder.OrthogonalLinks, OAI.Kadison.PartialIsometryOrder.familySpace and OAI.Kadison.PartialIsometryOrder.instCompleteSpaceSubtypeMemSubmoduleComplexFamilySpace (definitions bundle OAIQ039C001 of this split); OAI.Kadison.FactorCorner.instCompleteSpaceSubtypeMemSubmoduleComplexSupportSpace and OAI.Kadison.FactorCorner.supportSpace (definitions bundle OAIQ039C008 of this split).

Used by. The proof of OAI.Kadison.NormalRepresentation.exists_orthogonal_links_of_faithful of this split imports it.

Preamble

import Mathlib
import Definitions.Def_OAIQ039C001
import Definitions.Def_OAIQ039C008

Formal statement

section

set_option linter.all false
set_option linter.unusedVariables false
set_option linter.unusedTactic false
set_option linter.unreachableTactic false

namespace OAI

noncomputable section

section

noncomputable section

open scoped InnerProductSpace ComplexOrder Topology

namespace Kadison.PartialIsometryOrder

variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]

variable (M : VonNeumannAlgebra H) (e f : H →L[ℂ] H)

theorem maximal_family_spans_of_full_support
    (hsupport : (FactorCorner.supportSpace M e).starProjection=1)
    (heM : e∈M) (hfM : f∈M) (he : IsStarProjection e) (hf : IsStarProjection f)
    (S : Set (Link M e f)) (hS : Maximal (OrthogonalLinks M e f) S) :
    (familySpace M e f S).starProjection=f := by
  sorry

end Kadison.PartialIsometryOrder

end

end
end

end OAI
end

Source

OpenAI, Kadison's similarity theorem through uniform derivation estimates (result family 288), OpenAI Math Release, September 23, 2026, https://github.com/openai/math (result family 288; paper: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Kadisons-similarity-theorem-through-uniform-derivation-estimates-September-23-2026/paper.pdf; Apache License 2.0)

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