openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Operators /NormalAffiliatedTransport.lean
| import OAI.Analysis.IrrationalRotation.Model.NormalAffiliation | |
| import OAI.Analysis.IrrationalRotation.Operators.AffiliatedTransport | |
| import Mathlib.Analysis.CStarAlgebra.Unitary.Span | |
| namespace OAI | |
| noncomputable section | |
| namespace Invariant233 | |
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℂ E] | |
| [CompleteSpace E] | |
| theorem IsAffiliated.graph_mem_of_mem_commutant | |
| {M : VonNeumannAlgebra E} {H : E →ₗ.[ℂ] E} (hH : IsAffiliated M H) | |
| {W : E →L[ℂ] E} (hW : W ∈ M.commutant) (x : H.domain) : | |
| (W (x : E), W (H x)) ∈ H.graph := by | |
| let S : Submodule ℂ M.commutant := { | |
| carrier := {a | ∀ z : H.domain, | |
| ((a : E →L[ℂ] E) (z : E), (a : E →L[ℂ] E) (H z)) ∈ H.graph} | |
| zero_mem' := by | |
| intro z | |
| simp | |
| add_mem' := by | |
| intro a b ha hb z | |
| change ((a : E →L[ℂ] E) z + (b : E →L[ℂ] E) z, | |
| (a : E →L[ℂ] E) (H z) + (b : E →L[ℂ] E) (H z)) ∈ H.graph | |
| exact H.graph.add_mem (ha z) (hb z) | |
| smul_mem' := by | |
| intro c a ha z | |
| simpa only [SetLike.val_smul, smul_apply, Prod.smul_mk] | |
| using H.graph.smul_mem c (ha z) } | |
| have hspan : Submodule.span ℂ (unitary M.commutant : Set M.commutant) ≤ S := by | |
| apply Submodule.span_le.mpr | |
| intro U hU z | |
| let U' : unitary (E →L[ℂ] E) := | |
| ⟨(U : E →L[ℂ] E), Unitary.map_mem M.commutant.toStarSubalgebra.subtype hU⟩ | |
| obtain ⟨hD, hC⟩ := hH U' U.property | |
| exact H.mem_graph_iff.mpr ⟨⟨(U' : E →L[ℂ] E) z, hD z⟩, rfl, hC z⟩ | |
| rw [CStarAlgebra.span_unitary] at hspan | |
| exact hspan (Submodule.mem_top : (⟨W, hW⟩ : M.commutant) ∈ ⊤) x | |
| theorem IsAffiliated.commutes_of_mem_commutant | |
| {M : VonNeumannAlgebra E} {H : E →ₗ.[ℂ] E} (hH : IsAffiliated M H) | |
| {W : E →L[ℂ] E} (hW : W ∈ M.commutant) : | |
| ∃ hD : ∀ x : H.domain, W (x : E) ∈ H.domain, | |
| ∀ x : H.domain, H ⟨W (x : E), hD x⟩ = W (H x) := by | |
| let hD : ∀ x : H.domain, W (x : E) ∈ H.domain := fun x => | |
| H.mem_domain_of_mem_graph (hH.graph_mem_of_mem_commutant hW x) | |
| refine ⟨hD, fun x => ?_⟩ | |
| exact H.mem_graph_snd_inj (H.mem_graph ⟨W x, hD x⟩) | |
| (hH.graph_mem_of_mem_commutant hW x) rfl | |
| theorem IsAffiliated.transported_projection_mem | |
| {M : VonNeumannAlgebra E} {H : E →ₗ.[ℂ] E} (hH : IsAffiliated M H) | |
| {p : E →L[ℂ] E} (hpM : p ∈ M) (hp : IsStarProjection p) : | |
| (InvariantRotation.AffiliatedTransport.transported H p.range).starProjection ∈ M := by | |
| apply (VonNeumannAlgebra.IsStarProjection.mem_iff | |
| isStarProjection_starProjection M).mpr | |
| intro W hW | |
| rw [Submodule.range_starProjection] | |
| apply (Module.End.mem_invtSubmodule_iff_forall_mem_of_mem _).mpr | |
| obtain ⟨hD, hC⟩ := hH.commutes_of_mem_commutant hW | |
| have hpW := (VonNeumannAlgebra.IsStarProjection.mem_iff hp M).mp hpM W hW | |
| exact InvariantRotation.AffiliatedTransport.transported_invariant_of_commutes H p.range W | |
| ((Module.End.mem_invtSubmodule_iff_forall_mem_of_mem _).mp hpW) | |
| (fun x hx => hD ⟨x, hx⟩) hC | |
| theorem IsAffiliated.transported_projection_invariant | |
| {M : VonNeumannAlgebra E} {H : E →ₗ.[ℂ] E} (hH : IsAffiliated M H) | |
| (A B : E →L[ℂ] E) | |
| (hD : ∀ x : H.domain, A (x : E) ∈ H.domain) | |
| (hI : ∀ x : H.domain, H ⟨A (x : E), hD x⟩ = B (H x)) | |
| {p : E →L[ℂ] E} (hpM : p ∈ M) (hp : IsStarProjection p) | |
| (hinv : (1 - p) * A * p = 0) : | |
| let Q := (InvariantRotation.AffiliatedTransport.transported H p.range).starProjection | |
| Q ∈ M ∧ IsStarProjection Q ∧ (1 - Q) * B * Q = 0 := by | |
| dsimp only | |
| refine ⟨hH.transported_projection_mem hpM hp, isStarProjection_starProjection, ?_⟩ | |
| have hpap : p * A * p = A * p := by | |
| rw [sub_mul, one_mul, sub_mul, sub_eq_zero] at hinv | |
| exact hinv.symm | |
| have hpA : p.range ∈ Module.End.invtSubmodule A := by | |
| apply (ContinuousLinearMap.IsIdempotentElem.range_mem_invtSubmodule_iff | |
| hp.isIdempotentElem).mpr | |
| exact hpap | |
| have hQB := InvariantRotation.AffiliatedTransport.transported_invariant H p.range A B | |
| ((Module.End.mem_invtSubmodule_iff_forall_mem_of_mem _).mp hpA) | |
| (fun x hx => hD ⟨x, hx⟩) hI | |
| have hQinv : | |
| (InvariantRotation.AffiliatedTransport.transported H p.range).starProjection.range ∈ | |
| Module.End.invtSubmodule B := by | |
| rw [Submodule.range_starProjection] | |
| exact (Module.End.mem_invtSubmodule_iff_forall_mem_of_mem _).mpr hQB | |
| have hQBQ := (ContinuousLinearMap.IsIdempotentElem.range_mem_invtSubmodule_iff | |
| isStarProjection_starProjection.isIdempotentElem).mp hQinv | |
| simp only [← ContinuousLinearMap.mul_def] at hQBQ | |
| rw [sub_mul, one_mul, sub_mul, mul_assoc, hQBQ, sub_self] | |
| end Invariant233 | |
| end | |
| end OAI | |