| import Mathlib |
| import OAI.Analysis.FreeWords.PositiveMoments |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace KKFreeWords |
|
|
| open scoped BigOperators ComplexOrder |
|
|
| noncomputable section |
|
|
| variable {A : Type*} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A] |
|
|
| end |
|
|
| end KKFreeWords |
|
|
| namespace KKFreeWords |
|
|
| open scoped ComplexOrder |
|
|
| noncomputable section |
|
|
| variable {A : Type*} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A] |
|
|
| variable {τ : A →ₚ[ℂ] ℂ} {S P : StarSubalgebra ℂ A} |
|
|
| end |
|
|
| end KKFreeWords |
|
|
| namespace KKFreeWords |
|
|
| open scoped ComplexOrder |
|
|
| noncomputable section |
|
|
| variable {A : Type*} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A] |
|
|
| variable {ι : Type*} {τ : A →ₚ[ℂ] ℂ} {S : ι → StarSubalgebra ℂ A} |
|
|
| variable {i j : ι} |
|
|
| end |
|
|
| end KKFreeWords |
|
|
| namespace KKFreeWords |
|
|
| open scoped ComplexOrder BigOperators |
|
|
| noncomputable section |
|
|
| variable {A : Type*} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A] |
|
|
| end |
|
|
| end KKFreeWords |
|
|
| namespace KKFreeWords |
|
|
| open scoped BigOperators |
|
|
| noncomputable section |
|
|
| end |
|
|
| end KKFreeWords |
|
|
| namespace KKFreeWords |
|
|
| open scoped ComplexOrder BigOperators |
|
|
| noncomputable section |
|
|
| variable {A : Type*} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A] |
|
|
| variable {τ : A →ₚ[ℂ] ℂ} |
|
|
| end |
|
|
| end KKFreeWords |
|
|
| namespace KKFreeWords |
|
|
| open scoped ComplexOrder BigOperators |
|
|
| noncomputable section |
|
|
| variable {A : Type*} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A] |
|
|
| variable {τ : A →ₚ[ℂ] ℂ} |
|
|
| end |
|
|
| end KKFreeWords |
|
|
| namespace KKFreeWords |
|
|
| open scoped ComplexOrder BigOperators |
|
|
| noncomputable section |
|
|
| variable {A : Type*} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A] |
|
|
| variable {τ : A →ₚ[ℂ] ℂ} |
|
|
| end |
|
|
| end KKFreeWords |
|
|
| namespace KKFreeWords |
|
|
| open scoped ComplexOrder BigOperators |
|
|
| noncomputable section |
|
|
| variable {A : Type*} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A] |
|
|
| variable {τ : A →ₚ[ℂ] ℂ} |
|
|
| def cornerState (τ : A →ₚ[ℂ] ℂ) (n : ℕ) (p : A) : A →ₚ[ℂ] ℂ where |
| toFun x := (n:ℂ)*τ (star p*x*p) |
| map_add' x y := by simp only [mul_add,add_mul,map_add,mul_add] |
| map_smul' c x := by |
| simp only [mul_smul_comm,smul_mul_assoc,map_smul,smul_eq_mul,RingHom.id_apply] |
| ring |
| monotone' x y h := by |
| exact mul_le_mul_of_nonneg_left |
| (τ.monotone' (star_left_conjugate_le_conjugate h p)) (by positivity) |
|
|
| @[simp] theorem cornerState_apply (n : ℕ) (p x : A) : |
| cornerState τ n p x = (n:ℂ)*τ (star p*x*p) := rfl |
|
|
| theorem cornerState_one {n : ℕ} (hn : 0 < n) {p : A} |
| (hps : star p = p) (hpp : p*p = p) (htp : τ p = (n:ℂ)⁻¹) : |
| cornerState τ n p 1 = 1 := by |
| have hnz : (n:ℂ) ≠ 0 := Nat.cast_ne_zero.mpr hn.ne' |
| simp only [cornerState_apply,hps,mul_one,hpp,htp,mul_inv_cancel₀ hnz] |
| |
| theorem cornerState_supported (hτ : FaithfulTracial τ) (n : ℕ) {p x : A} |
| (hps : star p = p) (hpp : p*p = p) (hpx : p*x = x) : |
| cornerState τ n p x = (n:ℂ)*τ x := by |
| rw [cornerState_apply,hps,hτ.2.1 (p*x) p,← mul_assoc,hpp,hpx] |
| |
| theorem cornerState_leftMul_supported (hτ : FaithfulTracial τ) (n : ℕ) {p b : A} |
| (hps : star p = p) (hpp : p*p = p) (hpb : p*b = b) (x : A) : |
| cornerState τ n p (b*x) = (n:ℂ)*τ (b*x) := |
| cornerState_supported hτ n hps hpp (by rw [← mul_assoc,hpb]) |
| |
| omit [StarOrderedRing A] in |
| theorem dirichletRow_square_norm (hτ : FaithfulTracial τ) (u : unitary A) |
| {N : ℕ} (hN : 0 < N) : ‖(dirichletRow u N)^2‖ ≤ (N:ℝ) := by |
| let : Nontrivial A := nontrivial_of_ne (0:A) 1 (by |
| intro he |
| have ht := congrArg τ he |
| exact zero_ne_one (by simpa only [map_zero,hτ.1] using ht)) |
| have hNr : (0:ℝ) < N := by exact_mod_cast hN |
| have hr : ‖∑ r ∈ Finset.range N, upow u (r+1)‖ ≤ (N:ℝ) := by |
| calc _ ≤ ∑ r ∈ Finset.range N, ‖upow u (r+1)‖ := norm_sum_le _ _ |
| _ = _ := by simp [upow] |
| have hp := (norm_pow_le (∑ r ∈ Finset.range N, upow u (r+1)) 2).trans |
| (pow_le_pow_left₀ (norm_nonneg _) hr 2) |
| rw [dirichletRow_square u hN,norm_smul,Real.norm_eq_abs,abs_of_nonneg (inv_nonneg.mpr hNr.le)] |
| calc _ ≤ (N:ℝ)⁻¹ * (N:ℝ)^2 := mul_le_mul_of_nonneg_left hp (inv_nonneg.mpr hNr.le) |
| _ = _ := by field_simp |
| |
| theorem dirichletRow_fourth_upper (hτ : FaithfulTracial τ) (u : unitary A) |
| {N : ℕ} (hN : 0 < N) : |
| (τ (((dirichletRow u N)^2*star ((dirichletRow u N)^2))^2)).re ≤ |
| (N:ℝ)^2 * (τ ((dirichletRow u N)^2*star ((dirichletRow u N)^2))).re := by |
| exact (positiveMap_fourth_le ((dirichletRow u N)^2)).trans |
| (mul_le_mul_of_nonneg_right |
| (pow_le_pow_left₀ (norm_nonneg _) (dirichletRow_square_norm hτ u hN) 2) |
| (positiveMap_re_nonneg (mul_star_self_nonneg _))) |
| |
| end |
| |
| end KKFreeWords |
| end |
| |
| end OAI |
| |
