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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 | import Mathlib
import OAI.Analysis.Commutator.AtomicCoefficients
namespace OAI
noncomputable section
section
noncomputable section
open scoped ComplexOrder InnerProductSpace Topology BigOperators
open Filter
namespace Kadison.AtomicFactor.Decomposition
variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
variable {M : VonNeumannAlgebra H} {p : H →L[ℂ] H} (D : Decomposition M p)
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)
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]
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)]
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]
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
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]
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)
end Kadison.AtomicFactor.Decomposition
end
end
end
end OAI
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