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9eeb9c7 | 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 | import OAI.Analysis.IrrationalRotation.Operators.NormalTracialTransport
import OAI.Analysis.IrrationalRotation.Spectral.AffiliatedPositiveResolvent
import OAI.Analysis.IrrationalRotation.Spectral.PartialResolventCalculus
namespace OAI
noncomputable section
namespace Invariant233.NormalTraceDomainDensity
universe u
variable {E : Type u} [NormedAddCommGroup E] [InnerProductSpace ℂ E] [CompleteSpace E]
open NormalTracialTransport BoundedRangeTrace
theorem eq_of_le_of_trace_eq
(M : VonNeumannAlgebra E) (τ : TracialState M)
(hτ : IsNormalFaithfulTracialState M τ)
(p q : Submodule ℂ E) [CompleteSpace p] [CompleteSpace q]
(hp : p.starProjection ∈ M) (hq : q.starProjection ∈ M)
(hpq : p ≤ q)
(heq : extendedTrace M τ p.starProjection = extendedTrace M τ q.starProjection) :
p = q := by
have hproj : IsStarProjection (q.starProjection - p.starProjection) :=
isStarProjection_starProjection.sub_of_mul_eq_left isStarProjection_starProjection
(Submodule.starProjection_comp_starProjection_of_le hpq)
have hmem : q.starProjection - p.starProjection ∈ M := sub_mem hq hp
have hz : τ ⟨q.starProjection - p.starProjection, hmem⟩ = 0 := by
rw [← extendedTrace_apply M τ _ hmem, map_sub, heq, sub_self]
have hzero := hτ.1 ⟨q.starProjection - p.starProjection, hmem⟩
(ContinuousLinearMap.IsPositive.of_isStarProjection hproj) hz
have hpzero : q.starProjection - p.starProjection = 0 := congrArg Subtype.val hzero
have hprojection : q.starProjection = p.starProjection := sub_eq_zero.mp hpzero
have hrange := congrArg (fun A : E →L[ℂ] E => A.range) hprojection
simpa only [Submodule.range_starProjection] using hrange.symm
theorem closure_inf_eq_of_normalizer
(M : VonNeumannAlgebra E) (τ : TracialState M)
(hτ : IsNormalFaithfulTracialState M τ)
(A : E →L[ℂ] E) (hA : A ∈ M)
(hinj : Function.Injective A) (hadj : Function.Injective A.adjoint)
(d p : Submodule ℂ E) [CompleteSpace p]
(hD : A.range ≤ d) (hp : p.starProjection ∈ M) :
(p ⊓ d).topologicalClosure = p := by
let r := ((1 - p.starProjection) * A).ker
have hr : r.starProjection ∈ M := kernel_projection_mem M _
(mul_mem (sub_mem (one_mem M) hp) hA)
let q := (A * r.starProjection).range.topologicalClosure
let : IsClosed (q : Set E) := Submodule.isClosed_topologicalClosure _
let : CompleteSpace q := inferInstance
have hq : q.starProjection ∈ M := range_closure_projection_mem M _ (mul_mem hA hr)
have hrange : (A * r.starProjection).range ≤ p ⊓ d := by
rintro y ⟨x, rfl⟩
constructor
· have hx := r.starProjection_apply_mem x
change A (r.starProjection x) - p.starProjection (A (r.starProjection x)) = 0 at hx
exact p.starProjection_eq_self_iff.mp (sub_eq_zero.mp hx).symm
· exact hD ⟨r.starProjection x, rfl⟩
have hqp : q ≤ p := by
apply Submodule.topologicalClosure_minimal _ (hrange.trans inf_le_left)
exact (completeSpace_coe_iff_isComplete.mp inferInstance).isClosed
have htrace : extendedTrace M τ q.starProjection = extendedTrace M τ p.starProjection := by
exact (extendedTrace_range_eq_of_injective M τ hτ A hA hinj r hr).trans
(extendedTrace_preimage_eq M τ hτ A hA hadj p hp)
have hqeq : q = p := eq_of_le_of_trace_eq M τ hτ q p hq hp hqp htrace
apply le_antisymm
· apply Submodule.topologicalClosure_minimal _ inf_le_left
exact (completeSpace_coe_iff_isComplete.mp inferInstance).isClosed
· calc
p = q := hqeq.symm
_ ≤ (p ⊓ d).topologicalClosure := Submodule.topologicalClosure_mono hrange
theorem closure_inf_domain_eq_of_normalizer
(M : VonNeumannAlgebra E) (τ : TracialState M)
(hτ : IsNormalFaithfulTracialState M τ)
(H : E →ₗ.[ℂ] E) (A : E →L[ℂ] E) (hA : A ∈ M)
(hinj : Function.Injective A) (hadj : Function.Injective A.adjoint)
(hD : ∀ x, A x ∈ H.domain)
(p : Submodule ℂ E) [CompleteSpace p] (hp : p.starProjection ∈ M) :
(p ⊓ H.domain).topologicalClosure = p := by
apply closure_inf_eq_of_normalizer M τ hτ A hA hinj hadj H.domain p _ hp
rintro y ⟨x, rfl⟩
exact hD x
theorem closure_inf_domain_eq_of_affiliated
(M : VonNeumannAlgebra E) (τ : TracialState M)
(hτ : IsNormalFaithfulTracialState M τ)
(H : E →ₗ.[ℂ] E) (hH : IsSelfAdjoint H)
(hpos : IsPositivePartial H) (haff : IsAffiliated M H)
(p : Submodule ℂ E) [CompleteSpace p] (hp : p.starProjection ∈ M) :
(p ⊓ H.domain).topologicalClosure = p := by
obtain ⟨R, hRM, _, hD, hHR, _, _⟩ :=
PositivePartialResolvent.exists_affiliated_resolvent M H hH hpos haff
have hHR' : ∀ x : E, H ⟨R x, hD x⟩ = x - R x := fun x => hHR x (hD x)
have hinj : Function.Injective R := PartialResolvent.injective H R hD hHR'
have hformal : H.IsFormalAdjoint H := by
have hh := LinearPMap.adjoint_isFormalAdjoint hH.dense_domain
rwa [LinearPMap.isSelfAdjoint_def.mp hH] at hh
have hRself := PartialResolvent.isSelfAdjoint H R hD hHR' hformal
have hadj : Function.Injective R.adjoint := by
rw [← ContinuousLinearMap.star_eq_adjoint, hRself.star_eq]
exact hinj
exact closure_inf_domain_eq_of_normalizer M τ hτ H R hRM hinj hadj hD p hp
end Invariant233.NormalTraceDomainDensity
end
end OAI
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