| import Mathlib |
| import OAI.Analysis.Commutator.HilbertBiprodProjections |
| import OAI.Analysis.Commutator.HilbertFiniteMaps |
| import OAI.Analysis.Commutator.OrthogonalLinkSums |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| section |
|
|
| noncomputable section |
|
|
| open scoped ComplexOrder InnerProductSpace Topology |
|
|
| open Filter |
|
|
| namespace Kadison.BiprodLinkColumn |
|
|
| open HilbertBiprod PartialIsometryOrder HilbertFiniteMaps |
|
|
| variable {H G : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
| [NormedAddCommGroup G] [InnerProductSpace ℂ G] [CompleteSpace G] |
|
|
| variable (N : VonNeumannAlgebra (Space H G)) |
|
|
| abbrev Intertwiner := Link N (right (H:=H) (G:=G)) left |
|
|
| omit [CompleteSpace H] [CompleteSpace G] in |
| lemma inl_norm_le : ‖inlL (H:=H) (G:=G)‖≤1 := |
| (inl (H:=H) (G:=G)).norm_toContinuousLinearMap_le |
|
|
| omit [CompleteSpace H] [CompleteSpace G] in |
| lemma inr_norm_le : ‖inrL (H:=H) (G:=G)‖≤1 := |
| (inr (H:=H) (G:=G)).norm_toContinuousLinearMap_le |
|
|
| def component (v : Intertwiner N) : H →L[ℂ] G := |
| (inrL (H:=H) (G:=G)).adjoint.comp (v.val.adjoint.comp inlL) |
|
|
| lemma component_adjoint (v : Intertwiner N) : |
| (component N v).adjoint=(inlL (H:=H) (G:=G)).adjoint.comp (v.val.comp inrL) := by |
| simp only [component,ContinuousLinearMap.adjoint_comp,ContinuousLinearMap.adjoint_adjoint, |
| ContinuousLinearMap.comp_assoc] |
|
|
| lemma source_adjoint (v : Intertwiner N) : |
| (right (H:=H) (G:=G)).comp v.val.adjoint=v.val.adjoint := by |
| have hv : v.val.comp (right (H:=H) (G:=G))=v.val := by |
| simpa only [ContinuousLinearMap.mul_def] using v.mul_source right_projection |
| have h := congrArg (fun T : Space H G →L[ℂ] Space H G => T.adjoint) hv |
| rw [ContinuousLinearMap.adjoint_comp] at h |
| have hr : (right (H:=H) (G:=G)).adjoint=right := right_projection.isSelfAdjoint.star_eq |
| simpa only [hr] using h |
|
|
| lemma inr_component (v : Intertwiner N) : |
| (inrL (H:=H) (G:=G)).comp (component N v)=v.val.adjoint.comp inlL := by |
| rw [component,←ContinuousLinearMap.comp_assoc,inr_adjoint, |
| ←ContinuousLinearMap.comp_assoc,source_adjoint N v] |
|
|
| lemma component_square (v : Intertwiner N) : |
| (component N v).adjoint.comp (component N v)= |
| (inlL (H:=H) (G:=G)).adjoint.comp (v.final.comp inlL) := by |
| rw [component_adjoint,ContinuousLinearMap.comp_assoc,ContinuousLinearMap.comp_assoc, |
| inr_component] |
| rfl |
|
|
| variable (S : Set (Intertwiner N)) (hS : OrthogonalLinks N right left S) |
|
|
| def finiteColumn (s : Finset S) : H →L[ℂ] Copies G s := |
| column s (fun v => component N v.val.val) |
|
|
| def compression (s : Finset S) : H →L[ℂ] H := |
| (inlL (H:=H) (G:=G)).adjoint.comp ((finiteProjection N right left S s).comp inlL) |
|
|
| lemma column_square (s : Finset S) : |
| (finiteColumn N S s).adjoint.comp (finiteColumn N S s)=compression N S s := by |
| classical |
| rw [finiteColumn,adjoint_column_comp] |
| simp only [component_square] |
| ext x |
| simp [compression,finiteProjection,ContinuousLinearMap.comp_apply] |
| exact Finset.sum_attach s (fun v => (inlL (H:=H) (G:=G)).adjoint (v.val.final (inlL x))) |
|
|
| include hS |
|
|
| lemma compression_norm_le (s : Finset S) : ‖compression N S s‖≤1 := by |
| calc |
| _≤‖(inlL (H:=H) (G:=G)).adjoint‖* |
| (‖finiteProjection N right left S s‖*‖inlL (H:=H) (G:=G)‖) := |
| (ContinuousLinearMap.opNorm_comp_le _ _).trans |
| (mul_le_mul_of_nonneg_left (ContinuousLinearMap.opNorm_comp_le _ _) (norm_nonneg _)) |
| _≤1*(1*1) := mul_le_mul (by simpa only [ContinuousLinearMap.adjoint.norm_map] using inl_norm_le (H:=H) (G:=G)) |
| (mul_le_mul (finiteProjection_norm_le N right left S hS s) (inl_norm_le (H:=H) (G:=G)) |
| (norm_nonneg (inlL (H:=H) (G:=G))) (by norm_num)) |
| (mul_nonneg (norm_nonneg (finiteProjection N right left S s)) (norm_nonneg (inlL (H:=H) (G:=G)))) (by norm_num) |
| _=1 := by norm_num |
|
|
| lemma column_norm_le (s : Finset S) : ‖finiteColumn N S s‖≤1 := by |
| have h := compression_norm_le N S hS s |
| rw [←column_square N S s,ContinuousLinearMap.norm_adjoint_comp_self] at h |
| nlinarith [norm_nonneg (finiteColumn N S s)] |
|
|
| lemma compression_tendsto (hspan : (familySpace N right left S).starProjection=left) (x : H) : |
| Tendsto (fun s : Finset S => compression N S s x) atTop (𝓝 x) := by |
| have h := (inlL (H:=H) (G:=G)).adjoint.continuous.tendsto |
| (left (inlL (H:=H) (G:=G) x)) |
| have ht := h.comp (finiteProjection_tendsto N right left S hS left_projection hspan (inlL x)) |
| have he : (inlL (H:=H) (G:=G)).adjoint (left (inlL x))=x := by |
| rw [adjoint_inlL] |
| rfl |
| rw [he] at ht |
| exact ht |
|
|
| omit hS |
|
|
| lemma component_intertwines {A : Type*} [CStarAlgebra A] |
| (φ : A →⋆ₐ[ℂ] (H →L[ℂ] H)) (ψ : A →⋆ₐ[ℂ] (G →L[ℂ] G)) |
| (v : Intertwiner N) (a : A) |
| (hv : block (φ a) (ψ a)*star v.val=star v.val*block (φ a) (ψ a)) : |
| (component N v).comp (φ a)=(ψ a).comp (component N v) := by |
| ext x |
| have h := congrArg (fun T : Space H G →L[ℂ] Space H G => (T (inlL x)).ofLp.2) hv |
| simpa only [component,ContinuousLinearMap.comp_apply,adjoint_inrL,WithLp.sndL_apply, |
| mul_apply_eq_comp,ContinuousLinearMap.star_eq_adjoint,block_apply, |
| inlL_apply,WithLp.ofLp_toLp,WithLp.snd,map_zero] using h.symm |
|
|
| end Kadison.BiprodLinkColumn |
|
|
| end |
|
|
| end |
| end |
|
|
| end OAI |
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|
