| import Mathlib |
| import OAI.Analysis.TracialSplitting.PreparedBlocks |
| import OAI.Analysis.JiangSu.TensorProducts.TensorSliceSeparation |
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| namespace OAI |
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| noncomputable section |
| namespace MatrixTensor |
| open Pureness |
| open scoped TensorProduct CStarAlgebra ComplexOrder |
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| 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 _ _ |
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| theorem squareTensorMap_injective : Function.Injective (squareTensorMap p q r s) := by |
| intro z w h |
| exact sectionMap_injective _ _ (congrArg Subtype.val h) |
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| end JiangSuConstruction |
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| end |
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| end OAI |
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