| import Mathlib |
| import OAI.Analysis.Commutator.AtomicCoefficients |
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| namespace OAI |
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| noncomputable section |
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| section |
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| noncomputable section |
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| open scoped ComplexOrder InnerProductSpace Topology BigOperators |
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| open Filter |
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| namespace Kadison.AtomicFactor.Decomposition |
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| variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
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| variable {M : VonNeumannAlgebra H} {p : H →L[ℂ] H} (D : Decomposition M p) |
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| def finiteShift (f : D.I → D.I) (s : Finset D.I) : H →L[ℂ] H := |
| ∑ i∈s,D.v (f i)*star (D.v i) |
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| lemma finiteShift_star_mul (f : D.I → D.I) (hf : Function.Injective f) (s : Finset D.I) : |
| star (D.finiteShift f s)*D.finiteShift f s=D.support s := by |
| classical |
| rw [finiteShift,star_sum,Finset.sum_mul,support] |
| apply Finset.sum_congr rfl |
| intro i hi |
| rw [Finset.mul_sum,Finset.sum_eq_single_of_mem i hi] |
| · simp only [star_mul,star_star] |
| calc |
| _=(D.v i*(star (D.v (f i))*D.v (f i)))*star (D.v i) := by noncomm_ring |
| _=D.v i*star (D.v i) := by rw [D.initial,D.source] |
| · intro j hj hne |
| simp only [star_mul,star_star] |
| calc |
| _=(D.v i*(star (D.v (f i))*D.v (f j)))*star (D.v j) := by noncomm_ring |
| _=0 := by rw [D.cross (f i) (f j) (fun h => hne (hf h).symm),mul_zero,zero_mul] |
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| lemma finiteShift_norm_le (f : D.I → D.I) (hf : Function.Injective f) (s : Finset D.I) : |
| ‖D.finiteShift f s‖≤1 := by |
| have he := D.support_norm_le s |
| rw [←D.finiteShift_star_mul f hf s,CStarRing.norm_star_mul_self] at he |
| nlinarith [norm_nonneg (D.finiteShift f s)] |
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| lemma finiteShift_mul_val (f : D.I → D.I) (s : Finset D.I) {i : D.I} (hi : i∈s) : |
| D.finiteShift f s*D.v i=D.v (f i) := by |
| classical |
| rw [finiteShift,Finset.sum_mul,Finset.sum_eq_single_of_mem i hi,mul_assoc,D.initial,D.source] |
| intro j hj hne |
| rw [mul_assoc,D.cross j i hne,mul_zero] |
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| theorem exists_shift (f : D.I → D.I) (hf : Function.Injective f) : |
| ∃ W : H →L[ℂ] H,W∈M ∧ ∀ i,W*D.v i=D.v (f i) := by |
| classical |
| let ω : Ultrafilter (Finset D.I) := Ultrafilter.of atTop |
| have hω : (ω : Filter (Finset D.I))≤atTop := Ultrafilter.of_le _ |
| obtain ⟨W,hW,hlim⟩ := WeakOperatorLimit.exists_ultralimit ω (D.finiteShift f) 1 |
| (by norm_num) (D.finiteShift_norm_le f hf) |
| have hWM : W∈M := WeakOperatorLimit.mem_vonNeumann hlim M |
| (Eventually.of_forall fun s => M.sum_mem (fun i _ => M.mul_mem (D.mem (f i)) (M.star_mem' (D.mem i)))) |
| refine ⟨W,hWM,?_⟩ |
| intro i |
| ext x |
| apply ext_inner_right ℂ |
| intro y |
| apply tendsto_nhds_unique (hlim (D.v i x) y) |
| apply tendsto_const_nhds.congr' |
| filter_upwards [hω (eventually_ge_atTop ({i}:Finset D.I))] with s hs |
| have he := D.finiteShift_mul_val f s (i:=i) (hs (Finset.mem_singleton_self i)) |
| exact congrArg (fun T : H →L[ℂ] H => inner ℂ (T x) y) he.symm |
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| lemma shift_isometry (f : D.I → D.I) (hf : Function.Injective f) |
| {W : H →L[ℂ] H} (hW : ∀ i,W*D.v i=D.v (f i)) : star W*W=1 := by |
| apply D.eq_of_coefficients_eq |
| intro i j |
| calc |
| star (D.v i)*(star W*W)*D.v j=star (W*D.v i)*(W*D.v j) := by |
| rw [star_mul]; noncomm_ring |
| _=star (D.v (f i))*D.v (f j) := by rw [hW i,hW j] |
| _=star (D.v i)*D.v j := by |
| by_cases hij : i=j |
| · subst j; rw [D.initial,D.initial] |
| · rw [D.cross i j hij,D.cross (f i) (f j) (fun h => hij (hf h))] |
| _=star (D.v i)*1*D.v j := by rw [mul_one] |
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| lemma shifts_orthogonal (f g : D.I → D.I) (hfg : ∀ i j,f i≠g j) |
| {W V : H →L[ℂ] H} (hW : ∀ i,W*D.v i=D.v (f i)) |
| (hV : ∀ i,V*D.v i=D.v (g i)) : star W*V=0 := by |
| apply D.eq_zero_of_coefficients_zero |
| intro i j |
| calc |
| _=star (W*D.v i)*(V*D.v j) := by rw [star_mul]; noncomm_ring |
| _=star (D.v (f i))*D.v (g j) := by rw [hW i,hV j] |
| _=0 := D.cross _ _ (hfg i j) |
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| end Kadison.AtomicFactor.Decomposition |
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| end |
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| end |
| end |
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| end OAI |
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