File size: 5,218 Bytes
a97875e | 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 | 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
|