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ddc84c5 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 | 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
end OAI
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