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

Proved

OAI.Kadison.ColumnModule.hom_column_sq

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.ColumnModule, with open scoped Topology InnerProductSpace BoundedContinuousFunction, open Filter Set UniformSpace Completion):

theorem hom_column_sq {ι : Type*} [Fintype ι]
    (φ : A →ₐ[ℂ] (E →L[ℂ] E)) {M : ℝ} (hM : 0≤M)
    (hφ : ∀ a, ‖φ a‖≤ M*‖a‖) (a : ι → A) (ξ : E) :
    ∑ i, ‖φ (a i) ξ‖^2 ≤ M^4*‖ξ‖^2*‖∑ i,star (a i)*a i‖

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

Preamble

import Mathlib

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 Topology InnerProductSpace BoundedContinuousFunction

open Filter Set UniformSpace Completion

namespace Kadison.ColumnModule

variable {A E : Type*} [CStarAlgebra A]
  [NormedAddCommGroup E] [InnerProductSpace ℂ E]

theorem hom_column_sq {ι : Type*} [Fintype ι]
    (φ : A →ₐ[ℂ] (E →L[ℂ] E)) {M : ℝ} (hM : 0≤M)
    (hφ : ∀ a, ‖φ a‖≤ M*‖a‖) (a : ι → A) (ξ : E) :
    ∑ i, ‖φ (a i) ξ‖^2 ≤ M^4*‖ξ‖^2*‖∑ i,star (a i)*a i‖ := by
  sorry

end Kadison.ColumnModule

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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