| import OAI.Analysis.InterpolatedFactors.Ultraweak |
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| namespace OAI |
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| noncomputable section |
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| universe u_1 u_2 u_3 u_4 u_5 u_6 u_7 |
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| section |
| namespace FreeGroupFactorMain.FullReducingCorner |
| open Amplification GaussianRecognition ReducingSubspace |
| variable {A : Type u_1} {B : Type u_2} [CStarAlgebra A] [CStarAlgebra B] |
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| def transport (Φ : A ≃⋆ₐ[ℂ] B) (s : Set A) (t : Set B) (he : Φ '' s=t) : |
| bicomm s ≃⋆ₐ[ℂ] bicomm t where |
| toFun T := ⟨Φ T,by |
| rw [← he] |
| exact (bicommutant_transport Φ s T).mpr T.property⟩ |
| invFun T := ⟨Φ.symm T,by |
| apply (bicommutant_transport Φ s (Φ.symm T)).mp |
| rw [StarAlgEquiv.apply_symm_apply,he] |
| exact T.property⟩ |
| left_inv T := Subtype.ext (StarAlgEquiv.symm_apply_apply Φ (T:A)) |
| right_inv T := Subtype.ext (StarAlgEquiv.apply_symm_apply Φ (T:B)) |
| map_mul' T U := Subtype.ext (Φ.map_mul (T:A) (U:A)) |
| map_add' T U := Subtype.ext (Φ.map_add (T:A) (U:A)) |
| map_smul' c T := Subtype.ext (map_smul Φ c (T:A)) |
| map_star' T := Subtype.ext (Φ.map_star' (T:A)) |
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| variable {E : Type u_3} [NormedAddCommGroup E] [InnerProductSpace ℂ E] [CompleteSpace E] |
| variable (K : Submodule ℂ E) [CompleteSpace K] |
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| theorem representation_compress (T : E →L[ℂ] E) : |
| cornerRepresentation K (compress K.starProjection isStarProjection_starProjection T)=reduce K T := by |
| apply ContinuousLinearMap.ext |
| intro ξ |
| change K.orthogonalProjectionOnto (K.starProjection (T (K.starProjection (ξ:E))))= |
| K.orthogonalProjectionOnto (T ξ) |
| rw [K.starProjection_mem_subspace_eq_self ξ,K.orthogonalProjectionOnto_starProjection_of_le le_rfl] |
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| def reductionImage (s : Set (E →L[ℂ] E)) : Set (K →L[ℂ] K) := reduce K '' s |
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| theorem representation_compressedSet (s : Set (E →L[ℂ] E)) : |
| cornerRepresentation K '' compressedSet K.starProjection isStarProjection_starProjection s=reductionImage K s := by |
| rw [compressedSet,Set.image_image] |
| simp only [representation_compress,reductionImage] |
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| variable (s : Set (E →L[ℂ] E)) |
| variable (hps : K.starProjection∈StarSubalgebra.centralizer ℂ s) |
| variable (v : E →L[ℂ] E) (hvs : v∈StarSubalgebra.centralizer ℂ s) |
| variable (hpv : K.starProjection*v=v) (hvr : v*(1-K.starProjection)=v) (hvi : star v*v=1-K.starProjection) |
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| def restrictionIso : bicomm s ≃⋆ₐ[ℂ] bicomm (reductionImage K s) := |
| (iso K.starProjection isStarProjection_starProjection s hps v hvs hpv hvr hvi).trans |
| (transport (cornerRepresentation K) _ _ (representation_compressedSet K s)) |
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| theorem restrictionIso_apply (T : bicomm s) : |
| (restrictionIso K s hps v hvs hpv hvr hvi T : K →L[ℂ] K)=reduce K (T:E →L[ℂ] E) := |
| representation_compress K T |
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| @[instance_reducible] def operatorTopology {F : Type u_4} [NormedAddCommGroup F] |
| [InnerProductSpace ℂ F] [CompleteSpace F] (t : Set (F →L[ℂ] F)) : TopologicalSpace (bicomm t) := |
| TopologicalSpace.induced (fun T : bicomm t => (T:F →L[ℂ] F)) (HilbertUltraweak.topology F) |
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| theorem restrictionIso_normal : |
| @Continuous _ _ (operatorTopology s) (operatorTopology (reductionImage K s)) |
| (restrictionIso K s hps v hvs hpv hvr hvi) := by |
| let : TopologicalSpace (E →L[ℂ] E) := HilbertUltraweak.topology E |
| let : TopologicalSpace (bicomm s) := operatorTopology s |
| let : TopologicalSpace (K →L[ℂ] K) := HilbertUltraweak.topology K |
| apply continuous_induced_rng.mpr |
| have he : (fun T : bicomm s => (restrictionIso K s hps v hvs hpv hvr hvi T : K →L[ℂ] K))= |
| (fun T : bicomm s => reduce K (T:E →L[ℂ] E)) := funext (restrictionIso_apply K s hps v hvs hpv hvr hvi) |
| change @Continuous _ _ (operatorTopology s) (HilbertUltraweak.topology K) |
| (fun T : bicomm s => (restrictionIso K s hps v hvs hpv hvr hvi T : K →L[ℂ] K)) |
| rw [he] |
| exact (HilbertUltraweak.sandwich_continuous K.orthogonalProjectionOnto K.subtypeL).comp continuous_subtype_val |
| end FreeGroupFactorMain.FullReducingCorner |
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| namespace FreeGroupFactorMain.GaussianRecognition.System |
| open ReducingSubspace FullReducingCorner |
| variable {A : Type u_5} {E : Type u_6} {I : Type u_7} [Ring A] [StarRing A] [Algebra ℂ A] |
| variable [NormedAddCommGroup E] [InnerProductSpace ℂ E] [CompleteSpace E] |
| variable {covariance : I → I → A → A} (X : System A E I covariance) |
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| theorem normalHull_projection_mem (σ : I → I) : |
| X.normalHull.starProjection∈StarSubalgebra.centralizer ℂ (X.generators σ) := by |
| rw [StarSubalgebra.mem_centralizer_iff] |
| intro U hU |
| rcases hU with ⟨a,rfl⟩ | ⟨i,rfl⟩ |
| · exact ⟨(X.normalHull_projection_coeff a).eq.symm, |
| (by simpa only [map_star] using (X.normalHull_projection_coeff (star a)).eq.symm)⟩ |
| · refine ⟨(X.normalHull_projection_generator σ i).eq.symm,?_⟩ |
| change star (X.create i+star (X.create (σ i)))*_= _*star (X.create i+star (X.create (σ i))) |
| rw [star_add (R := Operator E),star_star] |
| exact ((X.normalHull_projection_annihilate i).add_right (X.normalHull_projection_create (σ i))).eq.symm |
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| theorem reduced_generators (σ : I → I) : |
| reductionImage X.normalHull (X.generators σ)=X.reducedSystem.generators σ := by |
| have hgen (i : I) : reduce X.normalHull (X.generator σ i)=X.reducedSystem.generator σ i := by |
| change reduce X.normalHull (X.create i+star (X.create (σ i)))= |
| reduce X.normalHull (X.create i)+star (reduce X.normalHull (X.create (σ i))) |
| rw [reduce_add,reduce_star] |
| change reduce X.normalHull '' (Set.range X.coeff ∪ Set.range (X.generator σ))= |
| Set.range X.reducedSystem.coeff ∪ Set.range (X.reducedSystem.generator σ) |
| rw [Set.image_union,← Set.range_comp,← Set.range_comp] |
| have hg : reduce X.normalHull ∘ X.generator σ=X.reducedSystem.generator σ := funext hgen |
| rw [hg] |
| rfl |
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| variable (σ : I → I) (v : E →L[ℂ] E) (hvs : v∈StarSubalgebra.centralizer ℂ (X.generators σ)) |
| variable (hpv : X.normalHull.starProjection*v=v) (hvr : v*(1-X.normalHull.starProjection)=v) |
| variable (hvi : star v*v=1-X.normalHull.starProjection) |
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| def fullReductionIso : X.generated σ ≃⋆ₐ[ℂ] X.reducedSystem.generated σ := by |
| have he := X.reduced_generators σ |
| exact (restrictionIso X.normalHull (X.generators σ) (X.normalHull_projection_mem σ) v hvs hpv hvr hvi).trans |
| (transport (StarAlgEquiv.refl ℂ _) _ _ (by simpa only [StarAlgEquiv.coe_refl,Set.image_id] using he)) |
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| theorem fullReductionIso_apply (T : X.generated σ) : |
| (X.fullReductionIso σ v hvs hpv hvr hvi T : X.normalHull →L[ℂ] X.normalHull)=reduce X.normalHull (T:Operator E) := |
| restrictionIso_apply X.normalHull _ _ v hvs hpv hvr hvi T |
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| theorem fullReductionIso_normal : |
| @Continuous _ _ (X.generatedTopology σ) (X.reducedSystem.generatedTopology σ) |
| (X.fullReductionIso σ v hvs hpv hvr hvi) := by |
| let : TopologicalSpace (Operator E) := ultraweak E |
| let : TopologicalSpace (X.generated σ) := X.generatedTopology σ |
| let : TopologicalSpace (X.normalHull →L[ℂ] X.normalHull) := ultraweak X.normalHull |
| apply continuous_induced_rng.mpr |
| have he : (fun T : X.generated σ => (X.fullReductionIso σ v hvs hpv hvr hvi T : X.normalHull →L[ℂ] X.normalHull))= |
| (fun T : X.generated σ => reduce X.normalHull (T:Operator E)) := funext (X.fullReductionIso_apply σ v hvs hpv hvr hvi) |
| change @Continuous _ _ (X.generatedTopology σ) (ultraweak X.normalHull) |
| (fun T : X.generated σ => (X.fullReductionIso σ v hvs hpv hvr hvi T : X.normalHull →L[ℂ] X.normalHull)) |
| rw [he] |
| exact (HilbertUltraweak.sandwich_continuous X.normalHull.orthogonalProjectionOnto X.normalHull.subtypeL).comp |
| continuous_subtype_val |
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| theorem fullReductionIso_trace (T : X.generated σ) : |
| inner ℂ X.reducedSystem.root ((X.fullReductionIso σ v hvs hpv hvr hvi T : X.normalHull →L[ℂ] X.normalHull) |
| X.reducedSystem.root)=inner ℂ X.root ((T:Operator E) X.root) := by |
| rw [fullReductionIso_apply] |
| exact inner_reduce X.normalHull _ _ _ |
| end FreeGroupFactorMain.GaussianRecognition.System |
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| end |
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| namespace FreeGroupFactorMain.WeightedFock |
| open FreeFock Amplification GaussianRecognition ReducingSubspace |
| variable (u v : ℝ) (hv : 0<v) (hvu : v<u) |
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| theorem weightedFull_left : |
| (weightedSystem u v hv hvu).normalHull.starProjection*pRightEdge u v=pRightEdge u v := by |
| rw [weightedHull_projection] |
| exact pRightEdge_left u v |
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| theorem weightedFull_right : |
| pRightEdge u v*(1-(weightedSystem u v hv hvu).normalHull.starProjection)=pRightEdge u v := by |
| rw [weightedHull_projection] |
| exact pRightEdge_right u v |
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| theorem weightedFull_initial : |
| star (pRightEdge u v)*pRightEdge u v=1-(weightedSystem u v hv hvu).normalHull.starProjection := by |
| rw [weightedHull_projection] |
| exact pRightEdge_initial u v hv hvu |
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| def fullGaussianIso : (weightedSystem u v hv hvu).generated scalarFlip ≃⋆ₐ[ℂ] |
| scalarSystem.generated scalarFlip := |
| ((weightedSystem u v hv hvu).fullReductionIso scalarFlip (pRightEdge u v) |
| (pRightEdge_mem_weightedCommutant u v hv hvu) (weightedFull_left u v hv hvu) |
| (weightedFull_right u v hv hvu) (weightedFull_initial u v hv hvu)).trans |
| (reducedGaussianIso u v hv hvu) |
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| theorem fullGaussianIso_normal : |
| @Continuous _ _ ((weightedSystem u v hv hvu).generatedTopology scalarFlip) |
| (scalarSystem.generatedTopology scalarFlip) (fullGaussianIso u v hv hvu) := by |
| let : TopologicalSpace ((weightedSystem u v hv hvu).generated scalarFlip) := |
| (weightedSystem u v hv hvu).generatedTopology scalarFlip |
| let : TopologicalSpace ((weightedSystem u v hv hvu).reducedSystem.generated scalarFlip) := |
| (weightedSystem u v hv hvu).reducedSystem.generatedTopology scalarFlip |
| let : TopologicalSpace (scalarSystem.generated scalarFlip) := scalarSystem.generatedTopology scalarFlip |
| exact (reducedGaussianIso_normal u v hv hvu).comp |
| ((weightedSystem u v hv hvu).fullReductionIso_normal scalarFlip (pRightEdge u v) |
| (pRightEdge_mem_weightedCommutant u v hv hvu) (weightedFull_left u v hv hvu) |
| (weightedFull_right u v hv hvu) (weightedFull_initial u v hv hvu)) |
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| theorem fullGaussianIso_trace (T : (weightedSystem u v hv hvu).generated scalarFlip) : |
| inner ℂ scalarSystem.root ((fullGaussianIso u v hv hvu T : FreeFock.Operator ScalarIndex) scalarSystem.root)= |
| inner ℂ pRoot ((T:PRange ℕ →L[ℂ] PRange ℕ) pRoot) := by |
| rw [fullGaussianIso,StarAlgEquiv.trans_apply,reducedGaussianIso_trace] |
| exact (weightedSystem u v hv hvu).fullReductionIso_trace scalarFlip (pRightEdge u v) |
| (pRightEdge_mem_weightedCommutant u v hv hvu) (weightedFull_left u v hv hvu) |
| (weightedFull_right u v hv hvu) (weightedFull_initial u v hv hvu) T |
| end FreeGroupFactorMain.WeightedFock |
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| end |
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| end OAI |
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