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a97875e | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 | import Mathlib
import OAI.Analysis.Commutator.AtomicDecomposition
import OAI.Analysis.Commutator.AtomicShift
namespace OAI
noncomputable section
section
noncomputable section
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 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
end Kadison.AtomicFactor.Decomposition
end
end
end
end OAI
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