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f1f7953 | 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 | import Mathlib
import OAI.Analysis.TracialSplitting.PreparedBlocks
import OAI.Analysis.JiangSu.TensorProducts.TensorSliceSeparation
namespace OAI
noncomputable section
namespace MatrixTensor
open Pureness
open scoped TensorProduct CStarAlgebra ComplexOrder
variable {I J : Type*} [Fintype I] [Fintype J] [DecidableEq I] [DecidableEq J]
instance matrixSubalgebraClosed (S : StarSubalgebra ℂ (Mat I)) :
IsClosed (S : Set (Mat I)) := S.toSubalgebra.toSubmodule.closed_of_finiteDimensional
instance matrixSubalgebraCStar (S : StarSubalgebra ℂ (Mat I)) : CStarAlgebra S :=
by
letI : SubringClass (StarSubalgebra ℂ (Mat I)) (Mat I) :=
@StarSubalgebra.subringClass ℂ (Mat I) _ _ _ _ _ _
exact StarSubalgebra.cstarAlgebra S
local instance (priority := 2000) matrixSubalgebraSemiring (S : StarSubalgebra ℂ (Mat I)) : Semiring S :=
(inferInstance : CStarAlgebra S).toSemiring
def fiberMap (S : StarSubalgebra ℂ (Mat I)) (T : StarSubalgebra ℂ (Mat J)) :
S ⊗[ℂ] T →⋆ₐ[ℂ] Mat (I×J) :=
commutingStarTensorHom (N := Mat (I×J)) (left.comp S.subtype) (right.comp T.subtype)
(fun (a : S) (b : T) => commute a.val b.val)
def fiberAlgebra (S : StarSubalgebra ℂ (Mat I)) (T : StarSubalgebra ℂ (Mat J)) :
StarSubalgebra ℂ (Mat (I×J)) := (fiberMap S T).range
lemma mul_mem_fiber (S : StarSubalgebra ℂ (Mat I)) (T : StarSubalgebra ℂ (Mat J))
{a : Mat I} {b : Mat J} (ha : a∈S) (hb : b∈T) :
left (J := J) a*right (I := I) b∈fiberAlgebra S T :=
⟨(⟨a,ha⟩:S) ⊗ₜ[ℂ] (⟨b,hb⟩:T),rfl⟩
lemma fiber_finite_sum (S : StarSubalgebra ℂ (Mat I)) (T : StarSubalgebra ℂ (Mat J))
{x : Mat (I×J)} (hx : x∈fiberAlgebra S T) :
∃ n : ℕ,∃ a : Fin n → S,∃ b : Fin n → T,
x=∑ i,left (a i).val*right (b i).val := by
obtain ⟨z,rfl⟩ := hx
obtain ⟨n,a,b,rfl⟩ := TensorProduct.exists_sum_tmul_eq z
refine ⟨n,a,b,?_⟩
rw [map_sum]
rfl
end MatrixTensor
namespace JiangSuConstruction
open MatrixTensor Pureness
open scoped CStarAlgebra TensorProduct ComplexOrder
variable (p q r s : ℕ)
local instance (priority := 2000) squareSectionSemiring :
Semiring C(Interval×Interval,MatrixTensor.Mat ((Fin p×Fin q)×(Fin r×Fin s))) :=
(inferInstance : Ring C(Interval×Interval,MatrixTensor.Mat ((Fin p×Fin q)×(Fin r×Fin s)))).toSemiring
def dropFiber (t : Interval) : StarSubalgebra ℂ (TensorMat p q) :=
if t=leftEndpoint then (leftAmplification p q).range
else if t=rightEndpoint then (rightAmplification p q).range else ⊤
lemma drop_mem_fiber (f : DimensionDrop p q) (t : Interval) : f.val t∈dropFiber p q t := by
unfold dropFiber
split_ifs with ht ht
· subst t; exact f.property.1
· subst t; exact f.property.2
· trivial
def squareSections : StarSubalgebra ℂ
C(Interval×Interval,MatrixTensor.Mat ((Fin p×Fin q)×(Fin r×Fin s))) where
carrier := {f | ∀ x y,f (x,y)∈fiberAlgebra (dropFiber p q x) (dropFiber r s y)}
one_mem' := by intro x y; exact one_mem _
zero_mem' := by intro x y; exact zero_mem _
add_mem' hf hg := by intro x y; exact add_mem (hf x y) (hg x y)
mul_mem' hf hg := by intro x y; exact mul_mem (hf x y) (hg x y)
algebraMap_mem' z := by intro x y; exact algebraMap_mem _ z
star_mem' hf := by intro x y; exact star_mem (hf x y)
instance squareSections_closed : IsClosed
(squareSections p q r s : Set C(Interval×Interval,MatrixTensor.Mat ((Fin p×Fin q)×(Fin r×Fin s)))) := by
change IsClosed {f : C(Interval×Interval,MatrixTensor.Mat ((Fin p×Fin q)×(Fin r×Fin s))) |
∀ x y,f (x,y)∈fiberAlgebra (dropFiber p q x) (dropFiber r s y)}
simp only [Set.ofPred_forall]
apply isClosed_iInter; intro x
apply isClosed_iInter; intro y
exact (matrixSubalgebraClosed _).preimage (continuous_eval_const (x,y))
abbrev Square := squareSections p q r s
lemma sectionMap_mem_square (z : DimensionDrop p q ⊗[ℂ] DimensionDrop r s) :
sectionMap (dimensionDrop p q) (dimensionDrop r s) z∈squareSections p q r s := by
induction z using TensorProduct.inductionOn with
| add z w hz hw =>
rw [map_add]
intro x y
exact (fiberAlgebra (dropFiber p q x) (dropFiber r s y)).add_mem (hz x y) (hw x y)
| tmul a b =>
intro x y
exact mul_mem_fiber _ _ (drop_mem_fiber p q a x) (drop_mem_fiber r s b y)
def squareTensorMap : DimensionDrop p q ⊗[ℂ] DimensionDrop r s →⋆ₐ[ℂ] Square p q r s where
toFun z := ⟨sectionMap (dimensionDrop p q) (dimensionDrop r s) z,sectionMap_mem_square p q r s z⟩
map_one' := by apply Subtype.ext; exact map_one _
map_zero' := by apply Subtype.ext; exact map_zero _
map_add' z w := by apply Subtype.ext; exact map_add _ _ _
map_mul' z w := by apply Subtype.ext; exact map_mul _ _ _
commutes' c := by apply Subtype.ext; exact (sectionMap (dimensionDrop p q) (dimensionDrop r s)).commutes c
map_star' z := by apply Subtype.ext; exact map_star _ _
theorem squareTensorMap_injective : Function.Injective (squareTensorMap p q r s) := by
intro z w h
exact sectionMap_injective _ _ (congrArg Subtype.val h)
end JiangSuConstruction
end
end OAI
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