openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Operators /CovariantTracialVector.lean
| import OAI.Analysis.IrrationalRotation.Model.OperatorProperties | |
| import OAI.Analysis.IrrationalRotation.Analysis.FieldVectorState | |
| import OAI.Analysis.IrrationalRotation.Operators.FieldTraceFaithfulness | |
| import OAI.Analysis.IrrationalRotation.Analysis.FieldPositivity | |
| import OAI.Analysis.IrrationalRotation.Operators.HilbertModelSize | |
| import OAI.Analysis.IrrationalRotation.Analysis.FieldCovariance | |
| namespace OAI | |
| open MeasureTheory Filter | |
| open scoped InnerProductSpace | |
| namespace Invariant233.FieldTraceCyclic | |
| variable {Ω E : Type*} [MeasurableSpace Ω] | |
| [NormedAddCommGroup E] [InnerProductSpace ℂ E] | |
| {μ : Measure Ω} | |
| theorem represented_eq_zero_of_field_ae_zero | |
| (T : Lp E 2 μ →L[ℂ] Lp E 2 μ) (A : Ω → E →L[ℂ] E) | |
| (hT : ∀ ξ : Lp E 2 μ, ∀ᵐ x ∂μ, (T ξ) x = A x (ξ x)) | |
| (hA : A =ᵐ[μ] 0) : T = 0 := by | |
| apply ContinuousLinearMap.ext | |
| intro ξ | |
| apply Lp.ext | |
| filter_upwards [hT ξ, hA, Lp.coeFn_zero E 2 μ] with x hx hxA hx0 | |
| simpa only [hxA, Pi.zero_apply, zero_apply, hx0] using hx | |
| variable [CompleteSpace E] [IsFiniteMeasure μ] | |
| theorem represented_eq_zero_of_vector_state_zero | |
| (b : HilbertBasis ℤ ℂ E) (T : Lp E 2 μ →L[ℂ] Lp E 2 μ) | |
| (A : Ω → E →L[ℂ] E) | |
| (hT : ∀ ξ : Lp E 2 μ, ∀ᵐ x ∂μ, (T ξ) x = A x (ξ x)) | |
| (hpos : ∀ᵐ x ∂μ, (A x).IsPositive) | |
| (σ : ℤ → Ω → Ω) (hσ : ∀ r, MeasurePreserving (σ r) μ μ) | |
| (hcov : ∀ r : ℤ, ∀ᵐ x ∂μ, | |
| inner ℂ (b 0) (A (σ r x) (b 0)) = inner ℂ (b r) (A x (b r))) | |
| (hzero : inner ℂ (Lp.const 2 μ (b 0)) (T (Lp.const 2 μ (b 0))) = 0) : | |
| T = 0 := by | |
| have hi : Integrable (fun x => inner ℂ (b 0) (A x (b 0))) μ := by | |
| apply (L2.integrable_inner (𝕜 := ℂ) (Lp.const 2 μ (b 0)) | |
| (T (Lp.const 2 μ (b 0)))).congr | |
| filter_upwards [Lp.coeFn_const 2 μ (b 0), hT (Lp.const 2 μ (b 0))] | |
| with x hx hTx | |
| rw [hTx, hx] | |
| rfl | |
| have heq : (fun x => (inner ℂ (A x (b 0)) (b 0)).re) = | |
| fun x => (inner ℂ (b 0) (A x (b 0))).re := by | |
| funext x | |
| exact inner_re_symm (𝕜 := ℂ) (A x (b 0)) (b 0) | |
| have hcov' : ∀ r : ℤ, ∀ᵐ x ∂μ, | |
| (inner ℂ (A x (b r)) (b r)).re = | |
| (inner ℂ (A (σ r x) (b 0)) (b 0)).re := by | |
| intro r | |
| filter_upwards [hcov r] with x hx | |
| calc | |
| (inner ℂ (A x (b r)) (b r)).re = (inner ℂ (b r) (A x (b r))).re := | |
| inner_re_symm (𝕜 := ℂ) (A x (b r)) (b r) | |
| _ = (inner ℂ (b 0) (A (σ r x) (b 0))).re := congrArg Complex.re hx.symm | |
| _ = (inner ℂ (A (σ r x) (b 0)) (b 0)).re := | |
| inner_re_symm (𝕜 := ℂ) (b 0) (A (σ r x) (b 0)) | |
| have hint : Integrable (fun x => (inner ℂ (A x (b 0)) (b 0)).re) μ := by | |
| rw [heq] | |
| exact hi.re | |
| have htr : (∫ x, (inner ℂ (A x (b 0)) (b 0)).re ∂μ) = 0 := by | |
| rw [heq] | |
| change (∫ x, RCLike.re (inner ℂ (b 0) (A x (b 0))) ∂μ) = 0 | |
| rw [integral_re hi, | |
| ← constant_vector_state_eq_integral T A hT (b 0), hzero] | |
| rfl | |
| exact represented_eq_zero_of_field_ae_zero T A hT | |
| (FieldTraceFaithfulness.ae_eq_zero_of_integral_diagonal_eq_zero | |
| μ b 0 A σ hσ hpos hcov' hint htr) | |
| theorem faithfulTracialVector_of_covariant_fields [IsProbabilityMeasure μ] | |
| (b : HilbertBasis ℤ ℂ E) (M : VonNeumannAlgebra (Lp E 2 μ)) | |
| (σ : ℤ → Ω → Ω) (hσ : ∀ r, MeasurePreserving (σ r) μ μ) | |
| (hfields : ∀ T : Lp E 2 μ →L[ℂ] Lp E 2 μ, T ∈ M → | |
| ∃ A : Ω → E →L[ℂ] E, | |
| (∀ ξ : Lp E 2 μ, ∀ᵐ x ∂μ, (T ξ) x = A x (ξ x)) ∧ | |
| (∀ r : ℤ, AEStronglyMeasurable (fun x => A x (b r)) μ) ∧ | |
| (∀ᵐ x ∂μ, ‖A x‖ ≤ ‖T‖) ∧ | |
| (∀ r k l : ℤ, ∀ᵐ x ∂μ, | |
| inner ℂ (b k) (A (σ r x) (b l)) = | |
| inner ℂ (b (k + r)) (A x (b (l + r)))) ∧ | |
| (T.IsPositive → ∀ᵐ x ∂μ, (A x).IsPositive)) : | |
| IsFaithfulTracialVector M (Lp.const 2 μ (b 0)) := by | |
| refine ⟨?_, ?_, ?_⟩ | |
| · rw [Lp.norm_const' 2 μ (b 0) (by norm_num) (by norm_num)] | |
| simp [b.orthonormal.norm_eq_one] | |
| · intro T S hTM hSM | |
| obtain ⟨A, hT, hA, hAbound, hcovA, _⟩ := hfields T hTM | |
| obtain ⟨B, hS, hB, hBbound, hcovB, _⟩ := hfields S hSM | |
| exact constant_vector_state_cyclic_of_covariance b T S A B hT hS hA hB | |
| ‖T‖ ‖S‖ (norm_nonneg _) (norm_nonneg _) hAbound hBbound σ hσ hcovA hcovB | |
| · intro T hTM hpos hzero | |
| obtain ⟨A, hT, _, _, hcov, hApos⟩ := hfields T hTM | |
| apply represented_eq_zero_of_vector_state_zero b T A hT (hApos hpos) σ hσ | |
| · intro r | |
| simpa only [zero_add] using hcov r 0 0 | |
| · exact hzero | |
| theorem faithfulTracialVector_of_represented_covariant_fields | |
| [IsProbabilityMeasure μ] | |
| (b : HilbertBasis ℤ ℂ E) (M : VonNeumannAlgebra (Lp E 2 μ)) | |
| (σ : ℤ → Ω → Ω) (hσ : ∀ r, MeasurePreserving (σ r) μ μ) | |
| (hfields : ∀ T : Lp E 2 μ →L[ℂ] Lp E 2 μ, T ∈ M → | |
| ∃ A : Ω → E →L[ℂ] E, | |
| (∀ ξ : Lp E 2 μ, ∀ᵐ x ∂μ, (T ξ) x = A x (ξ x)) ∧ | |
| (∀ r : ℤ, AEStronglyMeasurable (fun x => A x (b r)) μ) ∧ | |
| (∀ᵐ x ∂μ, ‖A x‖ ≤ ‖T‖) ∧ | |
| (∀ r k l : ℤ, ∀ᵐ x ∂μ, | |
| inner ℂ (b k) (A (σ r x) (b l)) = | |
| inner ℂ (b (k + r)) (A x (b (l + r))))) : | |
| IsFaithfulTracialVector M (Lp.const 2 μ (b 0)) := by | |
| let := HilbertModelSize.separable_of_countable_hilbertBasis b | |
| apply faithfulTracialVector_of_covariant_fields b M σ hσ | |
| intro T hTM | |
| obtain ⟨A, hT, hA, hAbound, hcov⟩ := hfields T hTM | |
| exact ⟨A, hT, hA, hAbound, hcov, | |
| ae_isPositive_of_represented_isPositive T A hT⟩ | |
| theorem faithfulTracialVector_of_commuting_shiftedPullbacks [IsProbabilityMeasure μ] | |
| (b : HilbertBasis ℤ ℂ E) (M : VonNeumannAlgebra (Lp E 2 μ)) | |
| (σ : ℤ → Ω → Ω) (hσ : ∀ r, MeasurePreserving (σ r) μ μ) | |
| (S : ℤ → E →ₗᵢ[ℂ] E) (hS : ∀ r k : ℤ, S r (b k) = b (k + r)) | |
| (hcomm : ∀ T : Lp E 2 μ →L[ℂ] Lp E 2 μ, T ∈ M → ∀ r, | |
| Commute T (DirectIntegral.shiftedPullback (S r).toContinuousLinearMap (σ r) (hσ r))) | |
| (hfields : ∀ T : Lp E 2 μ →L[ℂ] Lp E 2 μ, T ∈ M → | |
| ∃ A : Ω → E →L[ℂ] E, | |
| (∀ ξ : Lp E 2 μ, ∀ᵐ x ∂μ, (T ξ) x = A x (ξ x)) ∧ | |
| (∀ r : ℤ, AEStronglyMeasurable (fun x => A x (b r)) μ) ∧ | |
| (∀ᵐ x ∂μ, ‖A x‖ ≤ ‖T‖)) : | |
| IsFaithfulTracialVector M (Lp.const 2 μ (b 0)) := by | |
| apply faithfulTracialVector_of_represented_covariant_fields b M σ hσ | |
| intro T hTM | |
| obtain ⟨A, hT, hA, hAbound⟩ := hfields T hTM | |
| refine ⟨A, hT, hA, hAbound, ?_⟩ | |
| exact ae_matrix_covariance_of_commute b T A | |
| (fun r => DirectIntegral.shiftedPullback (S r).toContinuousLinearMap (σ r) (hσ r)) | |
| S σ hσ hS hT | |
| (fun r => shiftedPullback_ae (S r).toContinuousLinearMap (σ r) (hσ r)) (hcomm T hTM) | |
| end Invariant233.FieldTraceCyclic | |
| end OAI | |