| import Mathlib |
| import OAI.Analysis.Commutator.AtomicDecomposition |
| import OAI.Analysis.Commutator.AtomicShift |
|
|
| namespace OAI |
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| noncomputable section |
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| section |
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| noncomputable section |
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| namespace Kadison.AtomicFactor.Decomposition |
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| variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
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| variable {M : VonNeumannAlgebra H} {p : H →L[ℂ] H} (D : Decomposition M p) |
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| theorem two_isometries [Infinite D.I] : |
| ∃ s t : M,star s*s=1 ∧ star t*t=1 ∧ star s*t=0 := by |
| classical |
| obtain ⟨e⟩ : Nonempty (D.I ≃ D.I×Bool) := by |
| apply Cardinal.eq.mp |
| rw [Cardinal.mk_prod,Cardinal.mk_bool] |
| simpa using (Cardinal.mul_eq_left (Cardinal.aleph0_le_mk D.I) |
| ((Cardinal.natCast_le_aleph0 (n:=2)).trans (Cardinal.aleph0_le_mk D.I)) (by norm_num)).symm |
| let f : D.I → D.I := fun i => e.symm (i,false) |
| let g : D.I → D.I := fun i => e.symm (i,true) |
| have hf : Function.Injective f := by |
| intro i j h |
| exact congrArg Prod.fst (e.symm.injective h) |
| have hg : Function.Injective g := by |
| intro i j h |
| exact congrArg Prod.fst (e.symm.injective h) |
| have hfg : ∀ i j,f i≠g j := by |
| intro i j h |
| have he := congrArg Prod.snd (e.symm.injective h) |
| exact Bool.false_ne_true he |
| obtain ⟨W,hWM,hW⟩ := D.exists_shift f hf |
| obtain ⟨V,hVM,hV⟩ := D.exists_shift g hg |
| refine ⟨⟨W,hWM⟩,⟨V,hVM⟩,?_,?_,?_⟩ |
| · apply Subtype.ext |
| exact D.shift_isometry f hf hW |
| · apply Subtype.ext |
| exact D.shift_isometry g hg hV |
| · apply Subtype.ext |
| exact D.shifts_orthogonal f g hfg hW hV |
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| end Kadison.AtomicFactor.Decomposition |
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| end |
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| end |
| end |
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| end OAI |
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