| import OAI.Analysis.GroupDeterminants.ClosedTrace |
| |
| namespace OAI |
|
|
| universe uA |
|
|
| noncomputable section |
| open scoped BigOperators |
| |
| namespace GroupRingDeterminant |
|
|
| open Filter Topology |
|
|
| section WeightedPair |
| variable {A : Type uA} [Ring A] [Algebra ℝ A] |
| def weightedPair (Q R : A) (a b c : ℝ) : A := a • Q + b • R + c • (1 - Q - R) |
| lemma weightedPair_mul (Q R : A) (hQ : Q * Q = Q) (hR : R * R = R) |
| (hQR : Q * R = 0) (hRQ : R * Q = 0) (a b c d e f : ℝ) : |
| weightedPair Q R a b c * weightedPair Q R d e f = weightedPair Q R (a*d) (b*e) (c*f) := by |
| simp only [weightedPair, add_mul, mul_add, smul_mul_smul_comm, |
| mul_sub, sub_mul, one_mul, mul_one, hQ, hR, hQR, hRQ, smul_zero, |
| smul_sub] |
| module |
| lemma weightedPair_one (Q R : A) : weightedPair Q R 1 1 1 = 1 := by |
| simp only [weightedPair, one_smul] |
| lemma weightedPair_unit (Q R : A) (hQ : Q * Q = Q) (hR : R * R = R) |
| (hQR : Q * R = 0) (hRQ : R * Q = 0) (a b c : ℝ) |
| (ha : a ≠ 0) (hb : b ≠ 0) (hc : c ≠ 0) : IsUnit (weightedPair Q R a b c) := by |
| refine ⟨⟨weightedPair Q R a b c, weightedPair Q R a⁻¹ b⁻¹ c⁻¹, ?_, ?_⟩, rfl⟩ |
| · rw [weightedPair_mul Q R hQ hR hQR hRQ, mul_inv_cancel₀ ha, mul_inv_cancel₀ hb, |
| mul_inv_cancel₀ hc, weightedPair_one] |
| · rw [weightedPair_mul Q R hQ hR hQR hRQ, inv_mul_cancel₀ ha, inv_mul_cancel₀ hb, |
| inv_mul_cancel₀ hc, weightedPair_one] |
| lemma weightedPair_inverse (Q R : A) (hQ : Q * Q = Q) (hR : R * R = R) |
| (hQR : Q * R = 0) (hRQ : R * Q = 0) (a b c : ℝ) |
| (ha : a ≠ 0) (hb : b ≠ 0) (hc : c ≠ 0) : |
| Ring.inverse (weightedPair Q R a b c) = weightedPair Q R a⁻¹ b⁻¹ c⁻¹ := by |
| have hu := weightedPair_unit Q R hQ hR hQR hRQ a b c ha hb hc |
| apply_fun (weightedPair Q R a b c * ·) using IsUnit.mul_right_injective hu |
| dsimp only |
| rw [Ring.mul_inverse_cancel _ hu, weightedPair_mul Q R hQ hR hQR hRQ, |
| mul_inv_cancel₀ ha, mul_inv_cancel₀ hb, mul_inv_cancel₀ hc, weightedPair_one] |
| end WeightedPair |
|
|
| section PairTrace |
| variable {A : Type uA} [CStarAlgebra A] [Nontrivial A] |
|
|
| lemma trace_log_weightedPair (τ : A →L[ℝ] ℂ) |
| (hτ : ∀ a b, τ (a * b) = τ (b * a)) (hstar : ∀ a, τ (star a) = star (τ a)) |
| (Q R : A) (hQ : Q * Q = Q) (hR : R * R = R) |
| (hQR : Q * R = 0) (hRQ : R * Q = 0) (hQs : IsSelfAdjoint Q) (x : ℝ) : |
| τ (CFC.log (star (weightedPair Q R (Real.exp x) (-Real.exp (-x)) (-1)) * |
| weightedPair Q R (Real.exp x) (-Real.exp (-x)) (-1))) = |
| (x * (2 * (τ (Q - R)).re) : ℝ) := by |
| let M := fun y : ℝ => weightedPair Q R (Real.exp y) (-Real.exp (-y)) (-1) |
| let M' := fun y : ℝ => weightedPair Q R (Real.exp y) (Real.exp (-y)) 0 |
| have hd (y : ℝ) : HasDerivAt M (M' y) y := by |
| have h1 := (Real.hasDerivAt_exp y).smul_const Q |
| have h2 := ((Real.hasDerivAt_exp (-y)).comp y ((hasDerivAt_id y).neg)).neg.smul_const R |
| simpa [M, M', weightedPair] using (h1.add h2).add_const ((-1 : ℝ) • (1 - Q - R)) |
| have hdc : Continuous M' := by dsimp [M', weightedPair]; fun_prop |
| have hu (y : ℝ) : IsUnit (M y) := |
| weightedPair_unit Q R hQ hR hQR hRQ _ _ _ (Real.exp_ne_zero y) |
| (neg_ne_zero.mpr (Real.exp_ne_zero (-y))) (by norm_num) |
| have hi (y : ℝ) : M' y * Ring.inverse (M y) = Q - R := by |
| dsimp [M, M'] |
| rw [weightedPair_inverse Q R hQ hR hQR hRQ _ _ _ (Real.exp_ne_zero y) |
| (neg_ne_zero.mpr (Real.exp_ne_zero (-y))) (by norm_num), |
| weightedPair_mul Q R hQ hR hQR hRQ] |
| simp [weightedPair, Real.exp_ne_zero, sub_eq_add_neg] |
| have hdlog (y : ℝ) : HasDerivAt (fun y => τ (CFC.log (star (M y) * M y))) |
| (2 * (τ (Q - R)).re : ℝ) y := by |
| simpa only [hi y] using |
| hasDerivAt_trace_log_star_mul_self τ hτ hstar hd hdc.continuousAt hu (x := y) |
| have hdz (y : ℝ) : HasDerivAt (fun y => τ (CFC.log (star (M y) * M y)) - |
| (y * (2 * (τ (Q - R)).re) : ℝ)) 0 y := by |
| have hh := (((hasDerivAt_id y).mul_const (2 * (τ (Q - R)).re)).ofReal_comp) |
| simpa only [Pi.sub_def, id_eq, one_mul, sub_self] using (hdlog y).sub hh |
| have he := is_const_of_deriv_eq_zero (fun y => (hdz y).differentiableAt) |
| (fun y => (hdz y).deriv) x 0 |
| have hz : star (M 0) * M 0 = 1 := by |
| have he0 : M 0 = (2 : ℝ) • Q - 1 := by simp [M, weightedPair]; module |
| rw [he0] |
| simp only [star_sub, star_smul, star_one, hQs.star_eq, star_ofNat] |
| simp only [sub_mul, mul_sub, smul_mul_assoc, mul_smul_comm, |
| mul_one, one_mul, hQ] |
| module |
| simp only [hz, CFC.log_one, map_zero, zero_mul, Complex.ofReal_zero, sub_zero] at he |
| exact sub_eq_zero.mp he |
| |
| end PairTrace |
| |
| end GroupRingDeterminant |
| |
| end |
| |
| end OAI |
| |
