openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Operators /FullAffiliatedTransport.lean
| import OAI.Analysis.IrrationalRotation.Spectral.AffiliatedPositiveResolvent | |
| import OAI.Analysis.IrrationalRotation.Spectral.PartialResolventCalculus | |
| import OAI.Analysis.IrrationalRotation.Operators.NormalTracialTransport | |
| namespace OAI | |
| noncomputable section | |
| namespace Invariant233 | |
| theorem affiliated_transport_proof | |
| {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℂ E] [CompleteSpace E] | |
| (M : VonNeumannAlgebra E) (τ : TracialState M) | |
| (hτ : IsNormalFaithfulTracialState M τ) | |
| (H : E →ₗ.[ℂ] E) (hH : IsSelfAdjoint H) (hpos : IsPositivePartial H) | |
| (hinj : Function.Injective H) (haff : IsAffiliated M H) | |
| (A B : M) | |
| (hD : ∀ x : H.domain, (A : E →L[ℂ] E) (x : E) ∈ H.domain) | |
| (hinter : ∀ x : H.domain, | |
| H ⟨(A : E →L[ℂ] E) (x : E), hD x⟩ = (B : E →L[ℂ] E) (H x)) | |
| (p : M) (hp : IsStarProjection (p : E →L[ℂ] E)) | |
| (hinv : (1 - (p : E →L[ℂ] E)) * (A : E →L[ℂ] E) * | |
| (p : E →L[ℂ] E) = 0) : | |
| ∃ P : M, | |
| (P : E →L[ℂ] E) = | |
| (InvariantRotation.AffiliatedTransport.transported H | |
| (p : E →L[ℂ] E).range).starProjection ∧ | |
| IsStarProjection (P : E →L[ℂ] E) ∧ | |
| (1 - (P : E →L[ℂ] E)) * (B : E →L[ℂ] E) * | |
| (P : E →L[ℂ] E) = 0 ∧ | |
| τ P = τ p := by | |
| obtain ⟨hPM, hPproj, hPinv⟩ := haff.transported_projection_invariant | |
| (A : E →L[ℂ] E) (B : E →L[ℂ] E) hD hinter p.property hp hinv | |
| refine ⟨⟨_, hPM⟩, rfl, hPproj, hPinv, ?_⟩ | |
| obtain ⟨R, hRM, _hnorm, hRD, hHR, _hRH, _hrange⟩ := | |
| PositivePartialResolvent.exists_affiliated_resolvent M H hH hpos haff | |
| have hHR' : ∀ x, H ⟨R x, hRD x⟩ = x - R x := fun x => hHR x (hRD x) | |
| have hformal : H.IsFormalAdjoint H := by | |
| simpa only [LinearPMap.isSelfAdjoint_def.mp hH] using | |
| (LinearPMap.adjoint_isFormalAdjoint hH.dense_domain) | |
| have hRinj := PartialResolvent.injective H R hRD hHR' | |
| have hSinj := PartialResolvent.injective_one_sub H R hRD hHR' hinj | |
| have hRsa := PartialResolvent.isSelfAdjoint H R hRD hHR' hformal | |
| have hSsa := PartialResolvent.one_sub_isSelfAdjoint H R hRD hHR' hformal | |
| change star R = R at hRsa | |
| change star (1 - R : E →L[ℂ] E) = 1 - R at hSsa | |
| have hRadj : Function.Injective R.adjoint := by | |
| rw [← ContinuousLinearMap.star_eq_adjoint, hRsa] | |
| exact hRinj | |
| have hSadj : Function.Injective (1 - R : E →L[ℂ] E).adjoint := by | |
| rw [← ContinuousLinearMap.star_eq_adjoint, hSsa] | |
| exact hSinj | |
| have hSM : (1 - R : E →L[ℂ] E) ∈ M := sub_mem (one_mem M) hRM | |
| let : CompleteSpace (p : E →L[ℂ] E).range := | |
| (ContinuousLinearMap.IsIdempotentElem.isClosed_range hp.isIdempotentElem).completeSpace_coe | |
| have hpeq : (p : E →L[ℂ] E).range.starProjection = (p : E →L[ℂ] E) := | |
| (isStarProjection_iff_eq_starProjection_range.mp hp).choose_spec.symm | |
| have hpM : (p : E →L[ℂ] E).range.starProjection ∈ M := by | |
| rw [hpeq] | |
| exact p.property | |
| have htrace := NormalTracialTransport.transported_trace_eq_of_normalizer | |
| M τ hτ H hformal (LinearMap.ker_eq_bot.mpr hinj) R (1 - R) hRM hSM | |
| hRinj hRadj hSinj hSadj hRD hHR' (p : E →L[ℂ] E).range hpM hPM | |
| convert htrace using 1 | |
| congr 1 | |
| exact Subtype.ext hpeq.symm | |
| end Invariant233 | |
| end | |
| end OAI | |