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

Proved
OAI.Kadison.AtomicFactor.Decomposition.finite_dimensional

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.AtomicFactor.Decomposition, with open scoped ComplexOrder InnerProductSpace):

theorem finite_dimensional [Finite D.I] (hp : IsMinimalProjection M p) :
    FiniteDimensional ℂ M

Definitions. The statement is written with OAI.Kadison.AtomicFactor.Decomposition and OAI.Kadison.AtomicFactor.IsMinimalProjection (definitions bundle OAIQ039C000 of this split); OAI.Kadison.VonNeumannCStar.isClosed and OAI.Kadison.VonNeumannCStar.smulMemClass (definitions bundle OAIQ039C002 of this split).

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

Preamble
import Mathlib
import Definitions.Def_OAIQ039C000
import Definitions.Def_OAIQ039C002
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 ComplexOrder InnerProductSpace

namespace Kadison.AtomicFactor.Decomposition

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

variable {M : VonNeumannAlgebra H} {p : H →L[ℂ] H} (D : Decomposition M p)

theorem finite_dimensional [Finite D.I] (hp : IsMinimalProjection M p) :
    FiniteDimensional ℂ M := by
  sorry

end Kadison.AtomicFactor.Decomposition

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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