openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Operators /InvariantTraceTransport.lean
| import OAI.Analysis.IrrationalRotation.Operators.NormalTraceProjectionRigidity | |
| namespace OAI | |
| noncomputable section | |
| namespace Invariant233 | |
| variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] | |
| [CompleteSpace H] | |
| theorem NormalTraceProjectionRigidity.hasOnlyTrivialInvariantProjections_smul_iff | |
| (M : VonNeumannAlgebra H) | |
| (T : H →L[ℂ] H) {c : ℂ} (hc : c ≠ 0) : | |
| HasOnlyTrivialInvariantProjections M (c • T) ↔ | |
| HasOnlyTrivialInvariantProjections M T := by | |
| have hinv (p : H →L[ℂ] H) : (1 - p) * (c • T) * p = 0 ↔ | |
| (1 - p) * T * p = 0 := by | |
| simp only [mul_smul_comm, smul_mul_assoc, smul_eq_zero, hc, false_or] | |
| constructor | |
| · intro h p hpm hp hT | |
| exact h p hpm hp ((hinv p).mpr hT) | |
| · intro h p hpm hp hT | |
| exact h p hpm hp ((hinv p).mp hT) | |
| theorem hasOnlyTrivialInvariantProjections_of_trace_transport | |
| (M : VonNeumannAlgebra H) (τ : TracialState M) | |
| (hτ : IsNormalFaithfulTracialState M τ) (A B : M) | |
| (hB : HasOnlyTrivialInvariantProjections M (B : H →L[ℂ] H)) | |
| (htransport : ∀ p : M, IsStarProjection (p : H →L[ℂ] H) → | |
| (1 - (p : H →L[ℂ] H)) * (A : H →L[ℂ] H) * (p : H →L[ℂ] H) = 0 → | |
| ∃ P : M, IsStarProjection (P : H →L[ℂ] H) ∧ | |
| (1 - (P : H →L[ℂ] H)) * (B : H →L[ℂ] H) * (P : H →L[ℂ] H) = 0 ∧ | |
| τ P = τ p) : | |
| HasOnlyTrivialInvariantProjections M (A : H →L[ℂ] H) := by | |
| intro p hpm hp hinv | |
| let p' : M := ⟨p, hpm⟩ | |
| obtain ⟨P, hP, hPinv, htrace⟩ := htransport p' hp hinv | |
| have htrivP := hB (P : H →L[ℂ] H) P.property hP hPinv | |
| have hnontriv := NormalTraceProjectionRigidity.nontrivial_iff_of_trace_eq | |
| τ hτ hp hP htrace.symm | |
| by_contra hpnontriv | |
| push Not at hpnontriv | |
| have hp' : p' ≠ 0 ∧ p' ≠ 1 := by | |
| constructor | |
| · intro h | |
| exact hpnontriv.1 (congrArg Subtype.val h) | |
| · intro h | |
| exact hpnontriv.2 (congrArg Subtype.val h) | |
| have hP' := hnontriv.mp hp' | |
| rcases htrivP with hz | ho | |
| · exact hP'.1 (Subtype.ext hz) | |
| · exact hP'.2 (Subtype.ext ho) | |
| end Invariant233 | |
| end | |
| end OAI | |