File size: 6,017 Bytes
ddc84c5 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 | import OAI.Analysis.Cuntz.Regularization
import OAI.Analysis.Cuntz.Squares
namespace OAI
noncomputable section
/-! Regularized dimension states and subadditivity. -/
section
open scoped Matrix CStarAlgebra
namespace Kaplansky
attribute [local instance] CStarAlgebra.spectralOrder CStarAlgebra.spectralOrderedRing
variable {A : Type*} [CStarAlgebra A]
namespace CuntzDimensionState
/-- A common smaller cutoff controls two independent cutoff thresholds. -/
theorem cutoff_value_add_le_regularized (d : CuntzDimensionState A)
(a b : CuntzPositiveMatrix A) {ε δ : ℝ} (hε : 0 < ε) (hδ : 0 < δ) :
d.value (a.cutoff ε) + d.value (b.cutoff δ) ≤ d.regularized (a.add b) := by
have hp := lt_min hε hδ
calc
_ ≤ d.value (a.cutoff (min ε δ)) + d.value (b.cutoff (min ε δ)) :=
add_le_add (d.cutoff_value_antitone a (min_le_left _ _))
(d.cutoff_value_antitone b (min_le_right _ _))
_ = d.value ((a.add b).cutoff (min ε δ)) := by rw [CuntzPositiveMatrix.cutoff_add, d.value_add]
_ ≤ _ := d.cutoff_value_le_regularized _ hp
theorem regularized_add (d : CuntzDimensionState A) (a b : CuntzPositiveMatrix A) :
d.regularized (a.add b) = d.regularized a + d.regularized b := by
apply le_antisymm
· apply d.regularized_le
intro ε hε
rw [CuntzPositiveMatrix.cutoff_add, d.value_add]
exact add_le_add (d.cutoff_value_le_regularized a hε) (d.cutoff_value_le_regularized b hε)
· apply (le_sub_iff_add_le).mp
apply d.regularized_le a
intro ε hε
apply (le_sub_iff_add_le).mpr
apply (le_sub_iff_add_le').mp
apply d.regularized_le b
intro δ hδ
apply (le_sub_iff_add_le').mpr
exact d.cutoff_value_add_le_regularized a b hε hδ
/-- The regularization descends to an additive, normalized monotone state
on the actual Cuntz monoid. -/
def regularize (d : CuntzDimensionState A) : CuntzDimensionState A where
state := {
toFun := Quotient.lift d.regularized (fun _ _ h => d.regularized_eq h)
map_zero' := d.regularized_zero
map_add' := by
intro a b
induction a using Quotient.inductionOn with | h a =>
induction b using Quotient.inductionOn with | h b =>
exact d.regularized_add a b }
monotone := by
intro a b h
induction a using Quotient.inductionOn with | h a =>
induction b using Quotient.inductionOn with | h b =>
exact d.regularized_mono ((cuntzClass_le a b).mp h)
normalized := by
change d.regularized (.unit 1) = 1
exact_mod_cast d.regularized_unit 1
@[simp] theorem regularize_value (d : CuntzDimensionState A) (a : CuntzPositiveMatrix A) :
d.regularize.value a = d.regularized a := rfl
theorem regularize_lowerSemicontinuous (d : CuntzDimensionState A) (n : ℕ) :
LowerSemicontinuous (fun a : {a : CStarMatrix (Fin n) (Fin n) A // 0 ≤ a} =>
d.regularize.value ⟨n, CStarMatrix.ofMatrix.symm a.val, a.property⟩) :=
d.regularized_lowerSemicontinuous n
theorem regularize_idempotent_value (d : CuntzDimensionState A) (a : CuntzPositiveMatrix A) :
d.regularize.regularize.value a = d.regularize.value a :=
d.regularized_idempotent a
/-- Square swapping for the regularization follows from the proved
Cuntz equivalence, even for unequal matrix sizes. -/
theorem regularized_rectangular_gram (d : CuntzDimensionState A) {n m : ℕ}
(v : Matrix (Fin n) (Fin m) A)
(hv : 0 ≤ CStarMatrix.ofMatrix (v*v.conjTranspose))
(hw : 0 ≤ CStarMatrix.ofMatrix (v.conjTranspose*v)) :
d.regularized ⟨n, v*v.conjTranspose, hv⟩ =
d.regularized ⟨m, v.conjTranspose*v, hw⟩ :=
d.regularized_eq (cuntzEquivalent_rectangular_gram v)
end CuntzDimensionState
end Kaplansky
end
open scoped Matrix CStarAlgebra
namespace Kaplansky
attribute [local instance] CStarAlgebra.spectralOrder CStarAlgebra.spectralOrderedRing
variable {A : Type*} [CStarAlgebra A]
/-- The row of two identities compresses the diagonal block to the ordinary sum. -/
theorem cuntzLE_sum_block {n : ℕ} (a b : Matrix (Fin n) (Fin n) A) :
CuntzLE (a+b) (cuntzBlock a b) := by
let v := Matrix.fromCols (1 : Matrix (Fin n) (Fin n) A) (1 : Matrix (Fin n) (Fin n) A)
have h : v * cuntzBlock a b * v.conjTranspose = a+b := by
simp [v, cuntzBlock, Matrix.fromCols_mul_fromBlocks,
Matrix.conjTranspose_fromCols_eq_fromRows_conjTranspose,
Matrix.fromCols_mul_fromRows]
simpa only [h] using CuntzLE.exact (cuntzBlock a b) v
namespace CuntzDimensionState
/-- Dimension is subadditive for positive sums, without a commutation hypothesis. -/
theorem value_sum_le (d : CuntzDimensionState A) {n : ℕ}
(a b : CStarMatrix (Fin n) (Fin n) A) (ha : 0 ≤ a) (hb : 0 ≤ b) :
d.value ⟨n, CStarMatrix.ofMatrix.symm (a+b), add_nonneg ha hb⟩ ≤
d.value ⟨n, CStarMatrix.ofMatrix.symm a, ha⟩ +
d.value ⟨n, CStarMatrix.ofMatrix.symm b, hb⟩ := by
let pa : CuntzPositiveMatrix A := ⟨n, CStarMatrix.ofMatrix.symm a, ha⟩
let pb : CuntzPositiveMatrix A := ⟨n, CStarMatrix.ofMatrix.symm b, hb⟩
rw [← d.value_add pa pb]
apply d.value_mono
exact (cuntzLE_sum_block _ _).trans (CuntzPositiveMatrix.add_equiv pa pb).symm.1
theorem regularized_sum_le (d : CuntzDimensionState A) {n : ℕ}
(a b : CStarMatrix (Fin n) (Fin n) A) (ha : 0 ≤ a) (hb : 0 ≤ b) :
d.regularized ⟨n, CStarMatrix.ofMatrix.symm (a+b), add_nonneg ha hb⟩ ≤
d.regularized ⟨n, CStarMatrix.ofMatrix.symm a, ha⟩ +
d.regularized ⟨n, CStarMatrix.ofMatrix.symm b, hb⟩ :=
d.regularize.value_sum_le a b ha hb
theorem regularized_orthogonal_add (d : CuntzDimensionState A) {n : ℕ}
(a b : CStarMatrix (Fin n) (Fin n) A) (ha : 0 ≤ a) (hb : 0 ≤ b) (hab : a*b=0) :
d.regularized ⟨n, CStarMatrix.ofMatrix.symm (a+b), add_nonneg ha hb⟩ =
d.regularized ⟨n, CStarMatrix.ofMatrix.symm a, ha⟩ +
d.regularized ⟨n, CStarMatrix.ofMatrix.symm b, hb⟩ :=
d.regularize.value_orthogonal_add n a b ha hb hab
end CuntzDimensionState
end Kaplansky
end
end OAI
|