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

Proved

OAI.Kadison.Decomposable.homomorphisms_ae_of_countable_relations

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.Decomposable, with open MeasureTheory Filter, open scoped Topology):

theorem homomorphisms_ae_of_countable_relations
    (S : StarSubalgebra ℚ A) [Countable S]
    (hd : DenseRange (fun a : S => (a:A))) (hI : Complex.I • (1:A)∈S)
    (T : A → Ω → H →L[ℂ] H)
    (hb : ∀ a,∀ᵐ x ∂μ,‖T a x‖≤‖a‖)
    (ha : ∀ a b,∀ᵐ x ∂μ,T (a+b) x=T a x+T b x)
    (hm : ∀ a b,∀ᵐ x ∂μ,T (a*b) x=T a x*T b x)
    (hst : ∀ a,∀ᵐ x ∂μ,T (star a) x=star (T a x))
    (hsc : ∀ z : ℂ,∀ᵐ x ∂μ,T (z • (1:A)) x=z • (1:H →L[ℂ] H)) :
    ∀ᵐ x ∂μ,∃ Φ : A →⋆ₐ[ℂ] (H →L[ℂ] H),∀ a : S,Φ (a:A)=T (a:A) x

Definitions. The statement is written with OAI.Kadison.Decomposable.instNormedAlgebraRat_solutions (definitions bundle OAIQ039C007 of this split); OAI.Kadison.Decomposable.instStarModuleRat_solutions (definitions bundle OAIQ039C023 of this split).

Used by. The proof of OAI.Kadison.Decomposable.exists_pointwise_starHom of this split imports it.

Preamble

import Mathlib
import Definitions.Def_OAIQ039C023

Formal statement

section

attribute [-instance] _root_.OAI.Kadison.ConcreteCorner.closedCarrier _root_.OAI.Kadison.CyclicColumns.instDecidableEq_solutions _root_.OAI.Kadison.VonNeumannCStar.isClosed _root_.OAI.Kadison.VonNeumannCStar.smulMemClass _root_.OAI.Kadison.VonNeumannCStar.starOrderedRing
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 MeasureTheory Filter

open scoped Topology

namespace Kadison.Decomposable

variable {Ω H A : Type} [MeasurableSpace Ω] (μ : Measure Ω)
  [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
  [TopologicalSpace.SeparableSpace H] [IsFiniteMeasure μ] [CStarAlgebra A]

attribute [local instance] OAI.Kadison.Decomposable.instNormedAlgebraRat_solutions
attribute [local instance] OAI.Kadison.Decomposable.instNormedAlgebraRatContinuousLinearMapComplexId_solutions
attribute [local instance] OAI.Kadison.Decomposable.instStarModuleRat_solutions
local instance instStarModuleRatContinuousLinearMapComplexId_solutions_r73cf57_1 : StarModule ℚ (H →L[ℂ] H) where
  star_smul r a := by
    change star ((r:ℂ) • a)=((star r:ℚ):ℂ) • star a
    simp
omit [TopologicalSpace.SeparableSpace H] [IsFiniteMeasure μ] in
theorem homomorphisms_ae_of_countable_relations
    (S : StarSubalgebra ℚ A) [Countable S]
    (hd : DenseRange (fun a : S => (a:A))) (hI : Complex.I • (1:A)∈S)
    (T : A → Ω → H →L[ℂ] H)
    (hb : ∀ a,∀ᵐ x ∂μ,‖T a x‖≤‖a‖)
    (ha : ∀ a b,∀ᵐ x ∂μ,T (a+b) x=T a x+T b x)
    (hm : ∀ a b,∀ᵐ x ∂μ,T (a*b) x=T a x*T b x)
    (hst : ∀ a,∀ᵐ x ∂μ,T (star a) x=star (T a x))
    (hsc : ∀ z : ℂ,∀ᵐ x ∂μ,T (z • (1:A)) x=z • (1:H →L[ℂ] H)) :
    ∀ᵐ x ∂μ,∃ Φ : A →⋆ₐ[ℂ] (H →L[ℂ] H),∀ a : S,Φ (a:A)=T (a:A) x := by
  sorry

end Kadison.Decomposable

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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