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import OAI.Analysis.Commutator.FreeMoments
namespace OAI
noncomputable section
section
noncomputable section
namespace Kadison.AlgebraFreeExpectation
open FreeMoments
variable {A : Type*} [Ring A] [Algebra ℂ A] [StarRing A] [StarModule ℂ A]
variable (τ : A →ₗ[ℂ] ℂ) (hτ1 : τ 1=1) {ι : Type*} (S : ι → StarSubalgebra ℂ A)
variable (hfree : Free S τ)
lemma centered_mem {i : ι} {a : A} (ha : a ∈ S i) :
a - τ a • (1 : A) ∈ S i :=
sub_mem ha (SMulMemClass.smul_mem _ (one_mem (S i)))
omit [StarRing A] [StarModule ℂ A] in
include hτ1 in
lemma centered_zero (a : A) : τ (a - τ a • (1 : A)) = 0 := by
simp [map_sub, map_smul, hτ1]
include hfree in
lemma free_two {i j : ι} (hij : i ≠ j) {a b : A} (ha : a ∈ S i) (hb : b ∈ S j)
(ha0 : τ a = 0) (hb0 : τ b = 0) :
τ (a*b) = 0 := by
let x : Letter S τ := ⟨i, a, ha, ha0⟩
let y : Letter S τ := ⟨j, b, hb, hb0⟩
have h := hfree [x, y] (by simpa [Reduced, List.isChain_cons_cons, x, y]) (by simp)
simpa [word, x, y] using h
include hfree in
lemma free_three {i j : ι} (hij : i ≠ j) {a b c : A}
(ha : a ∈ S i) (hb : b ∈ S j) (hc : c ∈ S i)
(ha0 : τ a = 0) (hb0 : τ b = 0) (hc0 : τ c = 0) :
τ (a*b*c) = 0 := by
let x : Letter S τ := ⟨i, a, ha, ha0⟩
let y : Letter S τ := ⟨j, b, hb, hb0⟩
let z : Letter S τ := ⟨i, c, hc, hc0⟩
have h := hfree [x, y, z]
(by simp [Reduced, List.isChain_cons_cons, x, y, z, hij, hij.symm]) (by simp)
simpa [word, x, y, z, mul_assoc] using h
include hfree in
lemma free_four {i j : ι} (hij : i ≠ j) {a b c d : A}
(ha : a ∈ S i) (hb : b ∈ S j) (hc : c ∈ S i) (hd : d ∈ S j)
(ha0 : τ a = 0) (hb0 : τ b = 0)
(hc0 : τ c = 0) (hd0 : τ d = 0) :
τ (a*b*c*d) = 0 := by
let x : Letter S τ := ⟨i, a, ha, ha0⟩
let y : Letter S τ := ⟨j, b, hb, hb0⟩
let z : Letter S τ := ⟨i, c, hc, hc0⟩
let w : Letter S τ := ⟨j, d, hd, hd0⟩
have h := hfree [x, y, z, w]
(by simp [Reduced, List.isChain_cons_cons, x, y, z, w, hij, hij.symm]) (by simp)
simpa [word, x, y, z, w, mul_assoc] using h
include hτ1 hfree in
lemma centered_mul_zero {i j : ι} (hij : i ≠ j) {a b : A}
(ha : a ∈ S i) (hb : b ∈ S j) (ha0 : τ a = 0) :
τ (a*b) = 0 := by
have h := free_two τ S hfree hij ha (centered_mem τ S hb) ha0 (centered_zero τ hτ1 b)
simpa [mul_sub, mul_smul_comm, map_sub, map_smul, ha0] using h
include hτ1 hfree in
lemma mul_centered_zero {i j : ι} (hij : i ≠ j) {a b : A}
(ha : a ∈ S i) (hb : b ∈ S j) (hb0 : τ b = 0) :
τ (a*b) = 0 := by
have h := free_two τ S hfree hij (centered_mem τ S ha) hb (centered_zero τ hτ1 a) hb0
simpa [sub_mul, smul_mul_assoc, map_sub, map_smul, hb0] using h
include hτ1 hfree in
lemma centered_middle_pairing {i j : ι} (hij : i ≠ j) {a b c : A}
(ha : a ∈ S i) (hb : b ∈ S j) (hc : c ∈ S j)
(ha0 : τ a = 0) (hb0 : τ b = 0) :
τ (c*a*b) = 0 := by
have h := free_three τ S hfree hij.symm (centered_mem τ S hc) ha hb
(centered_zero τ hτ1 c) ha0 hb0
have hab := free_two τ S hfree hij ha hb ha0 hb0
simpa [sub_mul, smul_mul_assoc, map_sub, map_smul, hab, mul_assoc] using h
include hτ1 hfree in
lemma centered_sandwich_pairing {i j : ι} (hij : i ≠ j) {a b c : A}
(ha : a ∈ S i) (hb : b ∈ S j) (hc : c ∈ S j)
(ha0 : τ a = 0) (hb0 : τ b = 0) :
τ (c*a*b*a) = 0 := by
have h := free_four τ S hfree hij.symm (centered_mem τ S hc) ha hb ha
(centered_zero τ hτ1 c) ha0 hb0 ha0
have hab := free_three τ S hfree hij ha hb ha ha0 hb0 ha0
simpa [sub_mul, smul_mul_assoc, map_sub, map_smul, hab] using h
include hτ1 hfree in
theorem free_compression_pairing {i j : ι} (hij : i ≠ j) {e b c : A}
(he : e ∈ S i) (hb : b ∈ S j) (hc : c ∈ S j) (hb0 : τ b = 0) :
τ (c * (e*b*e)) = (τ e)^2 * τ (c*b) := by
let a := e - τ e • (1 : A)
have ha : a ∈ S i := centered_mem τ S he
have ha0 : τ a = 0 := centered_zero τ hτ1 e
have h1 := centered_sandwich_pairing τ hτ1 S hfree hij ha hb hc ha0 hb0
have h2 := centered_middle_pairing τ hτ1 S hfree hij ha hb hc ha0 hb0
have h3 := mul_centered_zero τ hτ1 S hfree hij.symm (mul_mem hc hb) ha ha0
have hid : c * (e*b*e) = c*a*b*a + τ e • (c*a*b) +
τ e • (c*b*a) + ((τ e)^2) • (c*b) := by
dsimp [a]
simp only [sub_mul, mul_sub, smul_mul_assoc, mul_smul_comm, mul_one, smul_sub, smul_smul, pow_two]
noncomm_ring
rw [hid]
simp only [map_add, map_smul, h1, h2, h3, smul_eq_mul, mul_zero, zero_add]
end Kadison.AlgebraFreeExpectation
end
end
end
end OAI
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