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

Proved

OAI.Kadison.CyclicSpectral.stableUnitary_intertwines

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.CyclicSpectral, with open MeasureTheory, open scoped Topology InnerProductSpace, open InfiniteAmplification Decomposable):

lemma stableUnitary_intertwines (W : Space ℕ H ≃ₗᵢ[ℂ] Space ℕ G) (V : Lp ℂ 2 μ ≃ₗᵢ[ℂ] G)
    (S : H →L[ℂ] H) (T : G →L[ℂ] G) (f : C(Ω,ℂ))
    (hW : ∀ x,W (map S x)=map T (W x))
    (hV : ∀ x,V (scalarMultiplier μ f x)=T (V x)) (v : Space ℕ H) :
    stableUnitary μ W V (map S v)=scalarVectorMultiplier μ f (stableUnitary μ W V v)

Definitions. The statement is written with OAI.Kadison.CyclicSpectral.stableUnitary, OAI.Kadison.InfiniteAmplification.Space and OAI.Kadison.InfiniteAmplification.map (definitions bundle OAIQ039C006 of this split); OAI.Kadison.CyclicSpectral.scalarMultiplier (definitions bundle OAIQ039C033 of this split); OAI.Kadison.Decomposable.scalarVectorMultiplier (definitions bundle OAIQ039C035 of this split).

Used by. The proof of OAI.Kadison.CentralSpectral.exists_stable_center_spectral_model of this split imports it.

Preamble

import Mathlib
import Definitions.Def_OAIQ039C033
import Definitions.Def_OAIQ039C035

Formal statement

section

attribute [-instance] _root_.OAI.Kadison.ConcreteCorner.closedCarrier _root_.OAI.Kadison.CyclicColumns.instDecidableEq_solutions _root_.OAI.Kadison.VonNeumannCStar.cstarAlgebra _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

open scoped Topology InnerProductSpace

namespace Kadison.CyclicSpectral

open InfiniteAmplification Decomposable

variable {Ω H G : Type*} [MeasurableSpace Ω] [TopologicalSpace Ω] [CompactSpace Ω]
  [BorelSpace Ω] (μ : Measure Ω)
  [NormedAddCommGroup H] [InnerProductSpace ℂ H]
  [NormedAddCommGroup G] [InnerProductSpace ℂ G]

lemma stableUnitary_intertwines (W : Space ℕ H ≃ₗᵢ[ℂ] Space ℕ G) (V : Lp ℂ 2 μ ≃ₗᵢ[ℂ] G)
    (S : H →L[ℂ] H) (T : G →L[ℂ] G) (f : C(Ω,ℂ))
    (hW : ∀ x,W (map S x)=map T (W x))
    (hV : ∀ x,V (scalarMultiplier μ f x)=T (V x)) (v : Space ℕ H) :
    stableUnitary μ W V (map S v)=scalarVectorMultiplier μ f (stableUnitary μ W V v) := by
  sorry

end Kadison.CyclicSpectral

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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