File size: 3,887 Bytes
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 | import OAI.Analysis.IrrationalRotation.Operators.NormalTracialTransport
import OAI.Analysis.IrrationalRotation.Analysis.VonNeumannSupportGeometry
namespace OAI
noncomputable section
namespace Invariant233.NormalTracialMeet
open NormalTracialTransport
universe u
variable {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H]
[CompleteSpace H]
theorem extendedTrace_support_eq (M : VonNeumannAlgebra H) (τ : TracialState M)
(hτ : IsNormalFaithfulTracialState M τ) (A : H →L[ℂ] H) (hA : A ∈ M) :
extendedTrace M τ A.kerᗮ.starProjection =
extendedTrace M τ A.range.topologicalClosure.starProjection := by
rw [Submodule.starProjection_orthogonal',
VonNeumannSupportGeometry.range_closure_projection_eq, map_sub, map_sub]
congr 1
exact extendedTrace_kernel_eq_adjoint_kernel M τ hτ A hA
theorem trace_support_eq (M : VonNeumannAlgebra H) (τ : TracialState M)
(hτ : IsNormalFaithfulTracialState M τ) (A : H →L[ℂ] H) (hA : A ∈ M) :
τ ⟨A.kerᗮ.starProjection,
VonNeumannSupportGeometry.kernel_orthogonal_projection_mem M A hA⟩ =
τ ⟨A.range.topologicalClosure.starProjection,
VonNeumannSupportGeometry.range_closure_projection_mem M A hA⟩ := by
simpa only [extendedTrace_apply M τ _
(VonNeumannSupportGeometry.kernel_orthogonal_projection_mem M A hA),
extendedTrace_apply M τ _
(VonNeumannSupportGeometry.range_closure_projection_mem M A hA)] using
extendedTrace_support_eq M τ hτ A hA
private instance inf_complete (p q : Submodule ℂ H)
[CompleteSpace p] [CompleteSpace q] : CompleteSpace ↥(p ⊓ q) := by
have hp : IsClosed (p : Set H) :=
(completeSpace_coe_iff_isComplete.mp inferInstance).isClosed
have hq : IsClosed (q : Set H) :=
(completeSpace_coe_iff_isComplete.mp inferInstance).isClosed
exact (hp.inter hq).completeSpace_coe
theorem extendedTrace_meet_lower_bound
(M : VonNeumannAlgebra H) (τ : TracialState M)
(hτ : IsNormalFaithfulTracialState M τ)
(p q : Submodule ℂ H) [CompleteSpace p] [CompleteSpace q]
(hp : p.starProjection ∈ M) (hq : q.starProjection ∈ M) :
(extendedTrace M τ p.starProjection).re +
(extendedTrace M τ q.starProjection).re - 1 ≤
(extendedTrace M τ (p ⊓ q).starProjection).re := by
let A := (1 - q.starProjection) * p.starProjection
have hA : A ∈ M := mul_mem (sub_mem (one_mem M) hq) hp
have hqperp : qᗮ.starProjection ∈ M := by
rw [Submodule.starProjection_orthogonal']
exact sub_mem (one_mem M) hq
have hmono := extendedTrace_projection_mono M τ
A.range.topologicalClosure qᗮ
(VonNeumannSupportGeometry.range_closure_projection_mem M A hA) hqperp
(VonNeumannSupportGeometry.range_closure_complement_projection_le p q)
rw [← extendedTrace_support_eq M τ hτ A hA] at hmono
change (extendedTrace M τ
(((1 - q.starProjection) * p.starProjection).kerᗮ.starProjection)).re ≤ _ at hmono
rw [VonNeumannSupportGeometry.initial_support_complement_projection,
map_sub, Complex.sub_re, extendedTrace_orthogonal, Complex.sub_re,
Complex.one_re] at hmono
linarith
theorem trace_meet_lower_bound
(M : VonNeumannAlgebra H) (τ : TracialState M)
(hτ : IsNormalFaithfulTracialState M τ)
(p q : Submodule ℂ H) [CompleteSpace p] [CompleteSpace q]
(hp : p.starProjection ∈ M) (hq : q.starProjection ∈ M) :
(τ ⟨p.starProjection, hp⟩).re + (τ ⟨q.starProjection, hq⟩).re - 1 ≤
(τ ⟨(p ⊓ q).starProjection,
VonNeumannSupportGeometry.inf_projection_mem p q M hp hq⟩).re := by
simpa only [extendedTrace_apply M τ _ hp, extendedTrace_apply M τ _ hq,
extendedTrace_apply M τ _
(VonNeumannSupportGeometry.inf_projection_mem p q M hp hq)] using
extendedTrace_meet_lower_bound M τ hτ p q hp hq
end Invariant233.NormalTracialMeet
end
end OAI
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