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

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OAI.Kadison.ConcreteCorner.representation_injective

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.ConcreteCorner, with open scoped InnerProductSpace, open MatrixCornerHilbert ReducingRepresentation):

lemma representation_injective : Function.Injective (representation hp)

Definitions. The statement is written with OAI.Kadison.ConcreteCorner.Algebra (definitions bundle OAIQ039C003 of this split); OAI.Kadison.Corner, OAI.Kadison.instAlgebraComplexCorner, OAI.Kadison.instRingCorner, OAI.Kadison.instStarRingCorner, OAI.Kadison.ConcreteCorner.closedCarrier, OAI.Kadison.MatrixCornerHilbert.fixedSpace and OAI.Kadison.MatrixCornerHilbert.instCompleteSpaceSubtypeMemSubmoduleComplexFixedSpace (definitions bundle OAIQ039C004 of this split); OAI.Kadison.ConcreteCorner.inclusion (definitions bundle OAIQ039C020 of this split); OAI.Kadison.ConcreteCorner.representation (definitions bundle OAIQ039C032 of this split).

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

Preamble

import Mathlib
import Definitions.Def_OAIQ039C032

Formal statement

section

attribute [-instance] _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 scoped InnerProductSpace

namespace Kadison.ConcreteCorner

open MatrixCornerHilbert ReducingRepresentation

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

variable (M : VonNeumannAlgebra H)

variable {M} {p : Algebra M} (hp : IsStarProjection p)

lemma representation_injective : Function.Injective (representation hp) := by
  sorry

end Kadison.ConcreteCorner

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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