openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Operators /FaithfulProjectionTrace.lean
| import OAI.Analysis.IrrationalRotation.Model.OperatorProperties | |
| import OAI.Analysis.IrrationalRotation.Rotation.ProjectionDiagonalPropagation | |
| namespace OAI | |
| noncomputable section | |
| open scoped InnerProductSpace | |
| namespace Invariant233 | |
| variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] | |
| [CompleteSpace H] | |
| theorem strict_positive_trace_of_nonzero_projection | |
| (M : VonNeumannAlgebra H) (ξ : H) (htrace : IsFaithfulTracialVector M ξ) | |
| (P : H →L[ℂ] H) (hPm : P ∈ M) (hP : IsStarProjection P) (hP0 : P ≠ 0) : | |
| 0 < (vectorTrace ξ P).re := by | |
| have hnonneg := (InvariantProjection.projection_re_inner_bounds hP ξ).1 | |
| apply lt_of_le_of_ne hnonneg | |
| intro heq | |
| have hzero : P ξ = 0 := | |
| (InvariantProjection.projection_re_inner_eq_zero_iff hP ξ).mp heq.symm | |
| apply hP0 | |
| exact htrace.2.2 P hPm (ContinuousLinearMap.IsPositive.of_isStarProjection hP) | |
| (by simp only [vectorTrace, hzero, inner_zero_right]) | |
| theorem strict_trace_bounds_of_nontrivial_projection | |
| (M : VonNeumannAlgebra H) (ξ : H) (htrace : IsFaithfulTracialVector M ξ) | |
| (P : H →L[ℂ] H) (hPm : P ∈ M) (hP : IsStarProjection P) | |
| (hP0 : P ≠ 0) (hP1 : P ≠ 1) : | |
| 0 < (vectorTrace ξ P).re ∧ (vectorTrace ξ P).re < 1 := by | |
| refine ⟨strict_positive_trace_of_nonzero_projection M ξ htrace P hPm hP hP0, ?_⟩ | |
| have hcomp := strict_positive_trace_of_nonzero_projection M ξ htrace | |
| (1 - P) (M.sub_mem M.one_mem hPm) hP.one_sub | |
| (sub_ne_zero.mpr (Ne.symm hP1)) | |
| have hself : (inner ℂ ξ ξ).re = 1 := by | |
| calc | |
| (inner ℂ ξ ξ).re = ‖ξ‖ ^ 2 := inner_self_eq_norm_sq (𝕜 := ℂ) ξ | |
| _ = 1 := by rw [htrace.1]; norm_num | |
| simp only [vectorTrace, sub_apply, | |
| one_apply_eq_self, inner_sub_right, Complex.sub_re, | |
| hself] at hcomp ⊢ | |
| exact sub_pos.mp hcomp | |
| end Invariant233 | |
| end | |
| end OAI | |