File size: 2,312 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 | import Mathlib
import OAI.Analysis.Commutator.AmplifiedStrongLimit
import OAI.Analysis.Commutator.OperatorAmplification
import OAI.Analysis.Commutator.ProjectionBlocks
namespace OAI
noncomputable section
noncomputable section
open Filter
open scoped InnerProductSpace ComplexOrder Topology BigOperators
namespace Kadison.ProjectionBlocks
variable {H ι : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
variable (P : ι → H →L[ℂ] H) (hp : ∀ i,IsStarProjection (P i))
(ho : ∀ i j,i≠j → P i*P j=0) (h : ℕ)
include hp in
lemma diagonal_projection (i : ι) : IsStarProjection (diagonalAmplification h (P i)) := by
change IsStarProjection (OperatorAmplification.diagonal (H:=H) (Fin h) (P i))
exact (hp i).map (OperatorAmplification.diagonal (H:=H) (Fin h))
include ho in
lemma diagonal_orthogonal (i j : ι) (hij : i≠j) :
diagonalAmplification h (P i)*diagonalAmplification h (P j)=0 := by
change OperatorAmplification.diagonal (H:=H) (Fin h) (P i)*OperatorAmplification.diagonal (H:=H) (Fin h) (P j)=0
rw [←map_mul,ho i j hij,map_zero]
lemma finiteSum_diagonal (s : Finset ι) :
finiteSum (fun i => diagonalAmplification h (P i)) s=diagonalAmplification h (finiteSum P s) := by
change (∑ i∈s,OperatorAmplification.diagonal (H:=H) (Fin h) (P i))=
OperatorAmplification.diagonal (H:=H) (Fin h) (∑ i∈s,P i)
exact (map_sum _ _ _).symm
include hp ho in
lemma diagonal_space_projection :
(space (fun i => diagonalAmplification h (P i))).starProjection=
diagonalAmplification h (space P).starProjection := by
apply ContinuousLinearMap.ext
intro x
have h₁ := finiteSum_tendsto (fun i => diagonalAmplification h (P i))
(diagonal_projection P hp h) (diagonal_orthogonal P ho h) x
have h₂ := StrongContractionLimit.diagonal_tendsto (finiteSum P) (space P).starProjection
(finiteSum_tendsto P hp ho) h x
simp only [←finiteSum_diagonal P h] at h₂
exact tendsto_nhds_unique h₁ h₂
include hp ho in
lemma diagonal_full (hfull : (space P).starProjection=1) :
(space (fun i => diagonalAmplification h (P i))).starProjection=1 := by
rw [diagonal_space_projection P hp ho h,hfull]
exact map_one (OperatorAmplification.diagonal (H:=H) (Fin h))
end Kadison.ProjectionBlocks
end
end
end OAI
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