| 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 |
|
|
