| import Mathlib |
| import OAI.Analysis.FreeWords.ProjectionSubdivision |
|
|
| namespace OAI |
|
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| noncomputable section |
|
|
| namespace KKProjectionSupport |
|
|
| open KKTraceTransfer KKUltraTrace KKFactorSupport KKFreeLiftFrontier |
|
|
| open scoped ComplexOrder |
|
|
| noncomputable section |
|
|
| def BitWords : ℕ → Type |
| | 0 => Unit |
| | n+1 => BitWords n × Bool |
| |
| instance bitWordsFintype (n : ℕ) : Fintype (BitWords n) := by |
| induction n with |
| | zero => exact inferInstanceAs (Fintype Unit) |
| | succ n ih => exact @instFintypeProd _ _ ih inferInstance |
| |
| instance bitWordsDecEq (n : ℕ) : DecidableEq (BitWords n) := Classical.decEq _ |
| |
| variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
| |
| local instance dyadicGridConcreteOrder (M : VonNeumannAlgebra H) : PartialOrder (ConcreteAlgebra M) := concreteOrder M |
| |
| local instance dyadicGridConcretePreorder (M : VonNeumannAlgebra H) : Preorder (ConcreteAlgebra M) := (concreteOrder M).toPreorder |
| |
| local instance dyadicGridConcreteLE (M : VonNeumannAlgebra H) : LE (ConcreteAlgebra M) := (concreteOrder M).toLE |
| |
| local instance dyadicGridConcreteSOR (M : VonNeumannAlgebra H) : StarOrderedRing (ConcreteAlgebra M) := concreteSOR M |
| |
| structure DyadicGrid (M : VonNeumannAlgebra H) (τ : FaithfulNormalTrace M) (n : ℕ) where |
| p : BitWords n → ConcreteAlgebra M |
| proj : ∀ w, IsStarProjection (p w) |
| orth : ∀ v w, v≠w → p v*p w=0 |
| sum : ∑ w, p w=1 |
| trace : ∀ w, τ.map (p w)=(((1/2:ℝ)^n : ℝ):ℂ) |
|
|
| def gridZero (M : VonNeumannAlgebra H) (τ : FaithfulNormalTrace M) : DyadicGrid M τ 0 where |
| p _ := 1 |
| proj _ := IsStarProjection.one _ |
| orth v w h := by exact (h (by cases v; cases w; rfl)).elim |
| sum := by change ∑ _ : Unit, (1:ConcreteAlgebra M)=1; simp |
| trace w := by simp only [pow_zero,Complex.ofReal_one,τ.normalized] |
|
|
| theorem grid_refine (M : VonNeumannAlgebra H) (τ : FaithfulNormalTrace M) |
| (hD : IsDiffuse M) {n : ℕ} (G : DyadicGrid M τ n) : |
| ∃ G' : DyadicGrid M τ (n+1), ∀ w, |
| G'.p (w,true)+G'.p (w,false)=G.p w := by |
| classical |
| have ht : 0 ≤ (1/2:ℝ)^n := le_of_lt (pow_pos (by norm_num) _) |
| have hex (w : BitWords n) : ∃ q : ConcreteAlgebra M, IsStarProjection q ∧ q*G.p w=q ∧ |
| τ.map q=(((1/2:ℝ)^n/2 : ℝ):ℂ) := by |
| apply projection_with_trace M τ hD (G.p w) (G.proj w) _ (div_nonneg ht (by norm_num)) |
| rw [G.trace,Complex.ofReal_re] |
| linarith |
| choose q hq hqp hτq using hex |
| have hle (w) : q w ≤ G.p w := ((hq w).le_iff_mul_eq_left (G.proj w)).mpr (hqp w) |
| let f : BitWords (n+1) → ConcreteAlgebra M := fun w => if w.2 then q w.1 else G.p w.1-q w.1 |
| have hf (w : BitWords (n+1)) : IsStarProjection (f w) := by |
| rcases w with ⟨w,b⟩; cases b |
| · exact ((hq w).le_iff_sub (G.proj w)).mp (hle w) |
| · exact hq w |
| have hfp (w : BitWords (n+1)) : f w ≤ G.p w.1 := by |
| rcases w with ⟨w,b⟩; cases b |
| · exact sub_le_self _ (hq w).nonneg |
| · exact hle w |
| have hff (v w : BitWords (n+1)) (hvw : v≠w) : f v*f w=0 := by |
| by_cases hfst : v.1=w.1 |
| · rcases v with ⟨v,b⟩; rcases w with ⟨w,c⟩; dsimp at hfst; subst w |
| cases b <;> cases c |
| · exact (hvw rfl).elim |
| · change (G.p v-q v)*q v=0 |
| rw [sub_mul,((hq v).le_iff_mul_eq_right (G.proj v)).mp (hle v),(hq v).isIdempotentElem.eq,sub_self] |
| · change q v*(G.p v-q v)=0 |
| rw [mul_sub,hqp,(hq v).isIdempotentElem.eq,sub_self] |
| · exact (hvw rfl).elim |
| · calc |
| f v*f w = (f v*G.p v.1)*(G.p w.1*f w) := by |
| rw [((hf v).le_iff_mul_eq_left (G.proj _)).mp (hfp v),((hf w).le_iff_mul_eq_right (G.proj _)).mp (hfp w)] |
| _ = f v*(G.p v.1*G.p w.1)*f w := by simp only [mul_assoc] |
| _ = 0 := by rw [G.orth _ _ hfst,mul_zero,zero_mul] |
| have hfs (w : BitWords n) : f (w,true)+f (w,false)=G.p w := by dsimp [f]; abel |
| have hsum : ∑ w, f w=1 := by |
| change ∑ w : BitWords n × Bool, f w=1 |
| rw [Fintype.sum_prod_type] |
| calc |
| _ = ∑ w, G.p w := Finset.sum_congr rfl fun w _ => (Fintype.sum_bool _).trans (hfs w) |
| _ = 1 := G.sum |
| have hfτ (w : BitWords (n+1)) : τ.map (f w)=(((1/2:ℝ)^(n+1):ℝ):ℂ) := by |
| rcases w with ⟨w,b⟩ |
| have hr : (1/2:ℝ)^(n+1)=(1/2:ℝ)^n/2 := by rw [pow_succ]; ring |
| rw [hr] |
| cases b |
| · change τ.map (G.p w-q w)=_ |
| rw [map_sub,G.trace,hτq] |
| push_cast |
| ring |
| · exact hτq w |
| exact ⟨⟨f,hf,hff,hsum,hfτ⟩,hfs⟩ |
|
|
| def dyadicGrid (M : VonNeumannAlgebra H) (τ : FaithfulNormalTrace M) (hD : IsDiffuse M) : |
| (n : ℕ) → DyadicGrid M τ n |
| | 0 => gridZero M τ |
| | n+1 => (grid_refine M τ hD (dyadicGrid M τ hD n)).choose |
|
|
| theorem dyadicGrid_refine (M : VonNeumannAlgebra H) (τ : FaithfulNormalTrace M) |
| (hD : IsDiffuse M) (n : ℕ) (w : BitWords n) : |
| (dyadicGrid M τ hD (n+1)).p (w,true)+(dyadicGrid M τ hD (n+1)).p (w,false)= |
| (dyadicGrid M τ hD n).p w := |
| (grid_refine M τ hD (dyadicGrid M τ hD n)).choose_spec w |
|
|
| end |
|
|
| end KKProjectionSupport |
| end |
|
|
| end OAI |
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|
