| import OAI.Analysis.FiniteFactor.Fiber.Shift |
| import OAI.Analysis.FiniteFactor.CrossedProduct.FieldAlgebra |
|
|
| namespace OAI |
|
|
| namespace FiniteFactor |
| noncomputable section |
|
|
| /-- The diagonal of an orthogonal projection is the square norm of its column. -/ |
| theorem fiberProjection_diagonal_inner (P : SourceFiberHilbert →L[ℂ] SourceFiberHilbert) |
| (hp : IsStarProjection P) (k : ℤ) : |
| P (lp.single 2 k (1 : ℂ)) k = |
| inner ℂ (P (lp.single 2 k (1 : ℂ))) (P (lp.single 2 k (1 : ℂ))) := by |
| let e : SourceFiberHilbert := lp.single 2 k (1 : ℂ) |
| have hi : P (P e) = P e := congrArg (fun A : SourceFiberHilbert →L[ℂ] SourceFiberHilbert => A e) |
| hp.isIdempotentElem |
| calc |
| _ = inner ℂ e (P e) := by simp [e,lp.inner_single_left] |
| _ = inner ℂ e (star P (P e)) := by rw [hp.isSelfAdjoint.star_eq,hi] |
| _ = _ := by rw [ContinuousLinearMap.star_eq_adjoint,ContinuousLinearMap.adjoint_inner_right] |
|
|
| theorem fiberProjection_diagonal_zero_iff (P : SourceFiberHilbert →L[ℂ] SourceFiberHilbert) |
| (hp : IsStarProjection P) (k : ℤ) : |
| P (lp.single 2 k (1 : ℂ)) k = 0 ↔ P (lp.single 2 k (1 : ℂ)) = 0 := by |
| rw [fiberProjection_diagonal_inner P hp,inner_self_eq_zero] |
|
|
| theorem fiberProjection_diagonal_one_iff (P : SourceFiberHilbert →L[ℂ] SourceFiberHilbert) |
| (hp : IsStarProjection P) (k : ℤ) : |
| P (lp.single 2 k (1 : ℂ)) k = 1 ↔ P (lp.single 2 k (1 : ℂ)) = lp.single 2 k (1 : ℂ) := by |
| have he := fiberProjection_diagonal_zero_iff (1-P) hp.one_sub k |
| change (lp.single 2 k (1 : ℂ) : SourceFiberHilbert) k - P (lp.single 2 k (1 : ℂ)) k = 0 ↔ |
| lp.single 2 k (1 : ℂ) - P (lp.single 2 k (1 : ℂ)) = 0 at he |
| simp only [lp.single_apply,Pi.single_eq_same] at he |
| rw [sub_eq_zero,sub_eq_zero] at he |
| simpa only [eq_comm] using he |
|
|
| theorem fiberProjection_diagonal_bounds (P : SourceFiberHilbert →L[ℂ] SourceFiberHilbert) |
| (hp : IsStarProjection P) (k : ℤ) : |
| 0 ≤ (P (lp.single 2 k (1 : ℂ)) k).re ∧ |
| (P (lp.single 2 k (1 : ℂ)) k).re ≤ 1 ∧ |
| (P (lp.single 2 k (1 : ℂ)) k).im = 0 := by |
| have hr : (P (lp.single 2 k (1 : ℂ)) k).re = ‖P (lp.single 2 k (1 : ℂ))‖ ^ 2 := by |
| rw [fiberProjection_diagonal_inner P hp] |
| exact inner_self_eq_norm_sq (𝕜 := ℂ) _ |
| have hb : 1 - (P (lp.single 2 k (1 : ℂ)) k).re = ‖(1-P) (lp.single 2 k (1 : ℂ))‖ ^ 2 := by |
| have h := congrArg Complex.re (fiberProjection_diagonal_inner (1-P) hp.one_sub k) |
| change ((lp.single 2 k (1 : ℂ) : SourceFiberHilbert) k - P (lp.single 2 k (1 : ℂ)) k).re = _ at h |
| simpa only [lp.single_apply,Pi.single_eq_same,Complex.sub_re,Complex.one_re, show ∀ v : SourceFiberHilbert, (inner ℂ v v).re = ‖v‖^2 from inner_self_eq_norm_sq (𝕜 := ℂ)] using h |
| refine ⟨by rw [hr]; positivity,by nlinarith [sq_nonneg ‖(1-P) (lp.single 2 k (1 : ℂ))‖],?_⟩ |
| rw [fiberProjection_diagonal_inner P hp] |
| exact inner_self_im (𝕜 := ℂ) _ |
|
|
| /-- If a diagonal entry is zero, that coordinate annihilates the entire range. -/ |
| theorem fiberProjection_range_coordinate_zero (P : SourceFiberHilbert →L[ℂ] SourceFiberHilbert) |
| (hp : IsStarProjection P) (k : ℤ) (hz : P (lp.single 2 k (1 : ℂ)) k = 0) |
| (v : SourceFiberHilbert) : P v k = 0 := by |
| have hzv := (fiberProjection_diagonal_zero_iff P hp k).mp hz |
| have he : inner ℂ (lp.single 2 k (1 : ℂ)) (P v) = |
| inner ℂ (P (lp.single 2 k (1 : ℂ))) v := by |
| rw [← ContinuousLinearMap.adjoint_inner_right P (lp.single 2 k (1 : ℂ)) v, ← ContinuousLinearMap.star_eq_adjoint,hp.isSelfAdjoint.star_eq] |
| simpa [lp.inner_single_left,hzv] using he |
|
|
| /-- Positivity of the diagonal weight transfers zero backwards and one forwards. -/ |
| theorem fiberProjection_diagonal_implications |
| (P Q D : SourceFiberHilbert →L[ℂ] SourceFiberHilbert) |
| (hp : IsStarProjection P) (hq : IsStarProjection Q) |
| (hi : (1-Q)*D*P = 0) (a : ℂ) (ha : a ≠ 0) |
| (hd : ∀ v : SourceFiberHilbert, D v 0 = a * v 0) |
| (he : D (lp.single 2 0 (1 : ℂ)) = a • lp.single 2 0 (1 : ℂ)) : |
| (Q (lp.single 2 0 (1 : ℂ)) 0 = 0 → P (lp.single 2 0 (1 : ℂ)) 0 = 0) ∧ |
| (P (lp.single 2 0 (1 : ℂ)) 0 = 1 → Q (lp.single 2 0 (1 : ℂ)) 0 = 1) := by |
| let e : SourceFiberHilbert := lp.single 2 0 (1 : ℂ) |
| have hfix : Q (D (P e)) = D (P e) := by |
| have hh := congrArg (fun A : SourceFiberHilbert →L[ℂ] SourceFiberHilbert => A e) hi |
| change D (P e) - Q (D (P e)) = 0 at hh |
| exact (sub_eq_zero.mp hh).symm |
| constructor |
| · intro hz |
| have hcoord := congrArg (fun v : SourceFiberHilbert => v 0) hfix |
| rw [fiberProjection_range_coordinate_zero Q hq 0 hz,hd] at hcoord |
| exact (mul_eq_zero.mp hcoord.symm).resolve_left ha |
| · intro h1 |
| have hpfix := (fiberProjection_diagonal_one_iff P hp 0).mp h1 |
| rw [hpfix,he,map_smul] at hfix |
| have hqfix : Q e = e := (smul_right_injective _ ha) hfix |
| exact (fiberProjection_diagonal_one_iff Q hq 0).mpr hqfix |
|
|
| end |
| end FiniteFactor |
|
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| end OAI |
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