| import OAI.Analysis.FiniteFactor.Convolution |
| import OAI.Analysis.FiniteFactor.Fiber.ProjectionDiagonal |
|
|
| namespace OAI |
|
|
| namespace FiniteFactor |
| open scoped ComplexConjugate |
| noncomputable section |
|
|
| / |
| theorem invariant_transport_composition {R : Type*} [Ring R] (P Q D W : R) |
| (hD : (1-Q)*D*P = 0) (hW : (1-Q)*W*Q = 0) : |
| (1-Q)*W*D*P = 0 := by |
| have hd : D*P = Q*D*P := by |
| apply sub_eq_zero.mp |
| simpa only [sub_mul,one_mul] using hD |
| calc |
| _ = (1-Q)*W*(D*P) := by simp only [mul_assoc] |
| _ = (1-Q)*W*(Q*D*P) := by rw [hd] |
| _ = ((1-Q)*W*Q)*D*P := by simp only [mul_assoc] |
| _ = 0 := by rw [hW,zero_mul,zero_mul] |
|
|
| theorem fiber_complement_inner_zero |
| (Q : SourceFiberHilbert →L[ℂ] SourceFiberHilbert) (hQ : IsStarProjection Q) |
| (u v : SourceFiberHilbert) (hv : (1-Q) v = 0) : |
| inner ℂ ((1-Q) u) v = 0 := by |
| rw [← ContinuousLinearMap.adjoint_inner_right (1-Q) u v, |
| ← ContinuousLinearMap.star_eq_adjoint,hQ.one_sub.isSelfAdjoint.star_eq,hv,inner_zero_right] |
|
|
| / |
| power identities hold. No square summability of rescaled coefficients is assumed. -/ |
| theorem fiber_transport_convolution_zero |
| (P Q D W : SourceFiberHilbert →L[ℂ] SourceFiberHilbert) |
| (hQ : IsStarProjection Q) (F G : ℤ → ℝ) (hG : ∀ k, G k ≠ 0) (n : ℤ) |
| (hD : (1-Q)*D*P = 0) (hW : (1-Q)*W*Q = 0) |
| (hDa : ∀ (v : SourceFiberHilbert) (k : ℤ), D v k = (G k / F k : ℂ) * v k) |
| (hWa : ∀ (v : SourceFiberHilbert) (k : ℤ), W v (n+k) = (G (n+k) / G k : ℂ) * v k) : |
| (∑' k, leftCoefficient F (P (lp.single 2 0 (1 : ℂ))) k * |
| rightCoefficient G ((1-Q) (lp.single 2 0 (1 : ℂ))) (-n-k)) = 0 := by |
| let ξ := P (lp.single 2 0 (1 : ℂ)) |
| let γ := (1-Q) (lp.single 2 0 (1 : ℂ)) |
| let η := W (D ξ) |
| have hη : ∀ k, η (n+k) = ((G (n+k) / F k : ℝ) : ℂ) * ξ k := by |
| intro k |
| dsimp [η] |
| rw [hWa,hDa,Complex.ofReal_div] |
| have hg : (G k : ℂ) ≠ 0 := by exact_mod_cast hG k |
| field_simp |
| have hz : (1-Q) η = 0 := by |
| exact congrArg (fun A : SourceFiberHilbert →L[ℂ] SourceFiberHilbert => A (lp.single 2 0 (1 : ℂ))) |
| (invariant_transport_composition P Q D W hD hW) |
| simp only [cocycle_convolution_term_eq] |
| rw [weighted_shift_convolution_eq_inner ξ γ η (fun k => G (n+k)/F k) n hη] |
| exact fiber_complement_inner_zero Q hQ _ η hz |
|
|
| end |
| end FiniteFactor |
|
|
| end OAI |
|
|
