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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
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