openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Operators /NormalTraceDomainDensity.lean
| import OAI.Analysis.IrrationalRotation.Operators.NormalTracialTransport | |
| import OAI.Analysis.IrrationalRotation.Spectral.AffiliatedPositiveResolvent | |
| import OAI.Analysis.IrrationalRotation.Spectral.PartialResolventCalculus | |
| namespace OAI | |
| noncomputable section | |
| namespace Invariant233.NormalTraceDomainDensity | |
| universe u | |
| variable {E : Type u} [NormedAddCommGroup E] [InnerProductSpace ℂ E] [CompleteSpace E] | |
| open NormalTracialTransport BoundedRangeTrace | |
| theorem eq_of_le_of_trace_eq | |
| (M : VonNeumannAlgebra E) (τ : TracialState M) | |
| (hτ : IsNormalFaithfulTracialState M τ) | |
| (p q : Submodule ℂ E) [CompleteSpace p] [CompleteSpace q] | |
| (hp : p.starProjection ∈ M) (hq : q.starProjection ∈ M) | |
| (hpq : p ≤ q) | |
| (heq : extendedTrace M τ p.starProjection = extendedTrace M τ q.starProjection) : | |
| p = q := by | |
| have hproj : IsStarProjection (q.starProjection - p.starProjection) := | |
| isStarProjection_starProjection.sub_of_mul_eq_left isStarProjection_starProjection | |
| (Submodule.starProjection_comp_starProjection_of_le hpq) | |
| have hmem : q.starProjection - p.starProjection ∈ M := sub_mem hq hp | |
| have hz : τ ⟨q.starProjection - p.starProjection, hmem⟩ = 0 := by | |
| rw [← extendedTrace_apply M τ _ hmem, map_sub, heq, sub_self] | |
| have hzero := hτ.1 ⟨q.starProjection - p.starProjection, hmem⟩ | |
| (ContinuousLinearMap.IsPositive.of_isStarProjection hproj) hz | |
| have hpzero : q.starProjection - p.starProjection = 0 := congrArg Subtype.val hzero | |
| have hprojection : q.starProjection = p.starProjection := sub_eq_zero.mp hpzero | |
| have hrange := congrArg (fun A : E →L[ℂ] E => A.range) hprojection | |
| simpa only [Submodule.range_starProjection] using hrange.symm | |
| theorem closure_inf_eq_of_normalizer | |
| (M : VonNeumannAlgebra E) (τ : TracialState M) | |
| (hτ : IsNormalFaithfulTracialState M τ) | |
| (A : E →L[ℂ] E) (hA : A ∈ M) | |
| (hinj : Function.Injective A) (hadj : Function.Injective A.adjoint) | |
| (d p : Submodule ℂ E) [CompleteSpace p] | |
| (hD : A.range ≤ d) (hp : p.starProjection ∈ M) : | |
| (p ⊓ d).topologicalClosure = p := by | |
| let r := ((1 - p.starProjection) * A).ker | |
| have hr : r.starProjection ∈ M := kernel_projection_mem M _ | |
| (mul_mem (sub_mem (one_mem M) hp) hA) | |
| let q := (A * r.starProjection).range.topologicalClosure | |
| let : IsClosed (q : Set E) := Submodule.isClosed_topologicalClosure _ | |
| let : CompleteSpace q := inferInstance | |
| have hq : q.starProjection ∈ M := range_closure_projection_mem M _ (mul_mem hA hr) | |
| have hrange : (A * r.starProjection).range ≤ p ⊓ d := by | |
| rintro y ⟨x, rfl⟩ | |
| constructor | |
| · have hx := r.starProjection_apply_mem x | |
| change A (r.starProjection x) - p.starProjection (A (r.starProjection x)) = 0 at hx | |
| exact p.starProjection_eq_self_iff.mp (sub_eq_zero.mp hx).symm | |
| · exact hD ⟨r.starProjection x, rfl⟩ | |
| have hqp : q ≤ p := by | |
| apply Submodule.topologicalClosure_minimal _ (hrange.trans inf_le_left) | |
| exact (completeSpace_coe_iff_isComplete.mp inferInstance).isClosed | |
| have htrace : extendedTrace M τ q.starProjection = extendedTrace M τ p.starProjection := by | |
| exact (extendedTrace_range_eq_of_injective M τ hτ A hA hinj r hr).trans | |
| (extendedTrace_preimage_eq M τ hτ A hA hadj p hp) | |
| have hqeq : q = p := eq_of_le_of_trace_eq M τ hτ q p hq hp hqp htrace | |
| apply le_antisymm | |
| · apply Submodule.topologicalClosure_minimal _ inf_le_left | |
| exact (completeSpace_coe_iff_isComplete.mp inferInstance).isClosed | |
| · calc | |
| p = q := hqeq.symm | |
| _ ≤ (p ⊓ d).topologicalClosure := Submodule.topologicalClosure_mono hrange | |
| theorem closure_inf_domain_eq_of_normalizer | |
| (M : VonNeumannAlgebra E) (τ : TracialState M) | |
| (hτ : IsNormalFaithfulTracialState M τ) | |
| (H : E →ₗ.[ℂ] E) (A : E →L[ℂ] E) (hA : A ∈ M) | |
| (hinj : Function.Injective A) (hadj : Function.Injective A.adjoint) | |
| (hD : ∀ x, A x ∈ H.domain) | |
| (p : Submodule ℂ E) [CompleteSpace p] (hp : p.starProjection ∈ M) : | |
| (p ⊓ H.domain).topologicalClosure = p := by | |
| apply closure_inf_eq_of_normalizer M τ hτ A hA hinj hadj H.domain p _ hp | |
| rintro y ⟨x, rfl⟩ | |
| exact hD x | |
| theorem closure_inf_domain_eq_of_affiliated | |
| (M : VonNeumannAlgebra E) (τ : TracialState M) | |
| (hτ : IsNormalFaithfulTracialState M τ) | |
| (H : E →ₗ.[ℂ] E) (hH : IsSelfAdjoint H) | |
| (hpos : IsPositivePartial H) (haff : IsAffiliated M H) | |
| (p : Submodule ℂ E) [CompleteSpace p] (hp : p.starProjection ∈ M) : | |
| (p ⊓ H.domain).topologicalClosure = p := by | |
| obtain ⟨R, hRM, _, hD, hHR, _, _⟩ := | |
| PositivePartialResolvent.exists_affiliated_resolvent M H hH hpos haff | |
| have hHR' : ∀ x : E, H ⟨R x, hD x⟩ = x - R x := fun x => hHR x (hD x) | |
| have hinj : Function.Injective R := PartialResolvent.injective H R hD hHR' | |
| have hformal : H.IsFormalAdjoint H := by | |
| have hh := LinearPMap.adjoint_isFormalAdjoint hH.dense_domain | |
| rwa [LinearPMap.isSelfAdjoint_def.mp hH] at hh | |
| have hRself := PartialResolvent.isSelfAdjoint H R hD hHR' hformal | |
| have hadj : Function.Injective R.adjoint := by | |
| rw [← ContinuousLinearMap.star_eq_adjoint, hRself.star_eq] | |
| exact hinj | |
| exact closure_inf_domain_eq_of_normalizer M τ hτ H R hRM hinj hadj hD p hp | |
| end Invariant233.NormalTraceDomainDensity | |
| end | |
| end OAI | |